Research ArticleAdaptive Control of a New Chaotic Financial System with IntegerOrder and Fractional Order and Its IdenticalAdaptive Synchronization

Paul Yaovi Dousseh1 Cyrille Ainamon1 Clement Hodevewan Miwadinou 12

Adjimon Vincent Monwanou1 and Jean Bio Chabi Orou1

1Laboratoire de Mecaniques des Fluides de la Dynamique Non-lineaire et de la Modelisation des SystemesBiologiques (LMFDNMSB) Institut de Mathematiques et de Sciences Physiques (IMSP) Porto-Novo Benin2Departement de Physique ENS-Natitingou Universite des Sciences Technologies Ingenierie et Mathematiques (UNSTIM)Abomey Benin

Correspondence should be addressed to Clement Hodevewan Miwadinou clementmiwadinouimsp-uacorg

Received 19 January 2021 Revised 13 February 2021 Accepted 26 February 2021 Published 13 March 2021

Academic Editor Yi Qi

Copyright copy 2021 Paul Yaovi Dousseh et al is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In this paper adaptive control and adaptive synchronization of an integer and fractional order new financial system withunknown constant parameters are studied Based on Lyapunovrsquos stability theory an adaptive control law is designed to as-ymptotically stabilize the state variables of the system to the origin in integer and fractional order cases By the same theory anadaptive synchronization law is designed to perform the identical synchronization of the new financial system in the cases ofinteger and fractional order with unknown constant parameters Numerical simulations are carried out in order to show theefficiency of the theoretical results

1 Introduction

Fractional order derivatives are a subject over 300 years oldinitiated by Leibnizs letter to LrsquoHospital [1 2] and are ageneralization of integer order derivatives But their appli-cations in scientific fields are very recent and this is due tothe lack of their physical interpretation e differencebetween these fractional order derivatives and the integerorder derivatives is that fractional order derivatives have thememory that turns out to be very useful when it comes todescribing systems with memory and heredity properties Inthe literature several systems have been described usingfractional order derivatives we can cite the fractional orderLiu system [3] the fractional order financial system [4] thefractional order glucose-insulin regulatory system [5] thefractional order Chua system [6] etc Chaotic dynamicalsystems are first of all nonlinear systems depending on

several parameters and having an extreme sensitivity toinitial conditionsese systems are found inmany scientificfields including chemical physical [6] economic [4] orbiological [5]is has led researchers from various horizonsto take an interest in these types of systems and especiallythe control of the chaos which intervenes and the syn-chronization of these systems with integer and fractionalorder Chaos control in a dynamical system consists indesigning a control law which stabilizes the system as-ymptotically on one of these unstable fixed points In theliterature several methods have been proposed to achievethis goal We have among others the linear feedback control[7] adaptive control [8 9] sliding mode control [10]Lyapunov-based nonlinear control [11] adaptive slidingmode control [12] etc Recently for the stabilization ofdynamical systems different results have been obtained inthe literature in fields as diverse as varied For example see

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 5512094 15 pageshttpsdoiorg10115520215512094

[13] where authors proceeded to the stabilization of a classof chaotic systems when systems are subject to uncertaintyand external disturbance by a new uncertainty and distur-bance estimator- (UDE-) based control method In [14] anovel distributed consensus algorithm based on the inte-gration of sliding mode control scheme and (average) ADTmethod is proposed to solve consensus control problem inorder to guarantee the stability of the closed-loop systemAlso in [15] the finite horizon control for a broad class oflinear It 1113954ostochastic differential equations (SDEs) withinfinite Markovian jumps and(x u v)-dependent noise isdone e authors proposed the existence of the mixedcontrol a necessary and sufficient condition which isrepresented by the solution of a countably infinite set ofcoupled generalized difference Riccati equations (GDREs)

e synchronization of integer and fractional ordersystems has also been widely discussed in the literaturedue to its applications in the field of communication[17 18] for the secure transmission of informationSeveral approaches are used for the synchronization ofchaotic systems such as synchronization via nonlinearcontrol [19 20] synchronization via active control[7 21 22] and adaptive synchronization [8 9 23] eparticularity of adaptive control and adaptive synchro-nization is that these unlike other controllers which areused when the systemrsquos parameters are known are usedwhen the systemrsquos parameters are unknown

In the field of economics several models have beenproposed [24 25] e study of dynamic behavior andthe control of chaos in financial and economic systemshave also been approached in order to understand thedynamic behavior of these systems and stabilize them inorder to eliminate undesirable behavior [10 26ndash31] In2020 Liao et al [32] presented a new model to take intoaccount the interaction between the various state vari-ables of the system e numerical study of this modelrevealed that it presents complex dynamic behaviors suchas period doubling and chaos [32] It would therefore beinteresting to control the chaos in this new financialsystem in other words to eliminate the undesirablebehaviors of the system by considering the case whereconstant parameters of the system are unknown and alsoto carry out the identical adaptive synchronization of thisnew chaotic system It is in this context that this work ispart of which in order to control the chaotic behavior ofthe new financial system when the parameters are un-known an adaptive control law will be designed to sta-bilize asymptotically at the origin the state variables of theinteger and fractional order system e case of theadaptive synchronization of the new financial systemwith integer and fractional order will also be discussed

e organization of the rest of this paper is as followsin Section 2 some concepts on fractional calculus and thedescription of the new financial system with integer andfractional order are given e adaptive control of chaosin the new financial system based on Lyapunovrsquos stabilitytheory in the cases of integer order and incommensuratefractional order are done in Section 3 Section 4 deals withthe adaptive synchronization of the new financial system

in the cases of integer and incommensurate fractionalorder Finally the conclusion is discussed in Section 5

2 Some Fractional Calculus Concepts andModel Description

e arbitrary order derivative in other words the frac-tional order derivative is a generalization of the integerorder derivative or the classical derivative We generallymeet in the literature three definitions of fractional orderderivative [33] In this paper we will use the fractionalorder derivative in the sense of Caputo because with thisderivative the initial conditions take the same form aswhen the system is defined with integer order derivative

e fractional order derivative in the sense of Caputo (C)is defined by

Ca D

qt f(t)

1Γ(n minus 1)

1113946t

a(t minus τ)

nminus qminus 1f

(n)(τ)dτ n minus 1lt qlt n

(1)

where Γ(middot)is the gamma function and q is the order of thefractional derivative

e fractional order derivative in the sense of Caputo hasa certain number of properties defined as follows [33 34]

Property 1 Suppose that 0lt qlt 1 then

Dy(t) D1minus q

Dqy(t) (2)

in which D (ddt)

Property 2 When q 0

D0y(t) y(t) (3)

Property 3 As in the case of the integer order derivative thefractional order derivative in the sense of Caputo is a linearoperator

Dq(cx(t) + δy(t)) cD

qx(t) + δD

qy(t) (4)

in which c and δ are real constants

Property 4 As in the case of the integer order derivative thefractional order derivative in the sense of Caputo satisfies theadditive index law i e

Dq1D

q2y(t) Dq2D

q1y(t) Dq1+q2y(t) (5)

with some reasonable constraints on the functiony(t)In 2020 Liao et al [32] presented a new financial model

in order to take into account the interaction between theinterest rate x(t) the investment demand y(t) and the priceindex z(t) e system is defined as follows

2 Mathematical Problems in Engineering

dx

dt dz +(y minus e)x

dy

dt minus ky

2minus lx

2+ m

dz

dt minus cz minus δx minus ρy

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

where the parameters e k c m l ρ and δ are constantsIn [32] when e 03 k 002 c 1 m 1 l 01 ρ

005 andd 12 δ 1 and initial conditions (12 15 16)are considered system (6) exhibits a chaotic behavior asshown in Figures 1(a)ndash1(d)When the initial conditions (0205 and 06) are considered system (6) also presents achaotic behavior [32]

e generalization of system (6) ie the fractional orderversion of the new financial system is also considered in thisstudy Classical derivatives (integer order) are replaced byfractional order derivatives as follows

Dq1x dz +(y minus e)x

Dq2y minus ky

2minus lx

2+ m

Dq3z minus cz minus δx minus ρy

⎧⎪⎪⎨

⎪⎪⎩(7)

where qi isin (0 1) and Dqi (dqi dtqi )(i 1 2 3) Ifq1 q2 q3 q then system (7) is said to be a commen-surate order system otherwise it is said to be an incom-mensurate order system

e new fractional order financial system is chaotic whenthe values of the above parameters are considered the initialconditions (12 15 16) and the orders q1 1 q2 088

and q3 1 are considerede phase diagrams projected onto the phase planes

(x y) (x z) and the time histories of the state variablesx(t) and y(t) are shown in Figures 2(a)ndash2(d)

3 Adaptive Control of Chaos in a NewFinancial System

31 Integer Order Case In this part we will be interested inthe design of an adaptive control law in order to globallystabilize the new integer order financial system

311 Controller Design To control the chaos in system (6)adaptive controllers are added to it e new controlledfinancial system can therefore be written in the followingform

_x dz +(y minus e)x + u1

_y minus ky2

minus lx2

+ m + u2

_z minus cz minus δx minus ρy + u3

⎧⎪⎪⎨

⎪⎪⎩(8)

in which _x (dxdt) _y (dydt) and _z (dzdt) eui (i 1 2 3) are adaptive controllers which will be sub-sequently designed taking into account the state variables ofthe system and the estimation of the unknown constantparameters d e k l m c δ and ρ of the system

To allow the state variables of the system to convergeasymptotically to the origin we take the following adaptivecontrol functions

u1 minus yx minus 1113954dz + 1113954ex minus h1x

u2 1113954ky2

+ 1113954lx2

minus 1113954m minus h2y

u3 1113954cz + 1113954δx + 1113954ρy minus h3z

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(9)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimation of theunknown constant parameters d e k l m c δ and ρ re-spectively and hi (i 1 2 3) are positive constants

By replacing the control law (9) in system (8) we have

_x (d minus 1113954d)z minus (e minus 1113954e)x minus h1x

_y minus (k minus 1113954k)y2

minus (l minus 1113954l)x2

+(m minus 1113954m) minus h2y

_z minus (c minus 1113954c)z minus (δ minus 1113954δ)x minus (ρ minus 1113954ρ)y minus h3z

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(10)

Let us define the estimation error of unknown param-eters as follows

ed d minus 1113954d

ee e minus 1113954e

ek k minus 1113954k

el l minus 1113954l

em m minus 1113954m

ec c minus 1113954c

eδ δ minus 1113954δ

eρ ρ minus 1113954ρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

By replacing equation (11) in system (10) we have

_x edz minus eex minus h1x

_y minus eky2

minus elx2

+ em minus h2y

_z minus ecz minus eδx minus eρy minus h3z

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(12)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider the Lyapunov quadratic functiondefined as follows

V 12

x2

+ y2

+ z2

+ e2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(13)

which is a positive definite function on R11e derivative with respect to time of equation (13) gives

us_V x _x + y _y + z _z + ed _ed + ee _ee + ek _ek

+ el _el + em _em + ec _ec + eδ _eδ + eρ _eρ(14)

which specify that

Mathematical Problems in Engineering 3

_ed minus_1113954d

_ee minus _1113954e

_ek minus_1113954k _el minus

_1113954l _em minus _1113954m _ec minus _1113954c _eδ minus_1113954δ _eρ minus _1113954ρ

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(15)

Using system (12) and equation (15) equations (14)becomes

_V minus h1x2

minus h2y2

minus h3z2

+ ed(xz minus_1113954d)

+ ee minus x2

minus _1113954e1113872 1113873 + ek minus y3

minus_1113954k1113874 1113875

+ el minus yx2

minus_1113954l1113874 1113875 + em(y minus _1113954m)

+ ec minus z2

minus _1113954c1113872 1113873 + eδ(minus xz minus_1113954δ)

+ eρ(minus zy minus _1113954ρ)

(16)

From equation (16) we deduce that the estimated pa-rameters update law is

_d xz + h4ed

_e minus x2

+ h5ee

_k minus y

3+ h6ek

_l minus yx

2+ h7el

_m y + h8em

_c minus z2

+ h9ec

_δ minus xz + h10eδ_ρ minus zy + h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

in which h4 h5 h6 h7 h8 h9 h10 and h11 are positiveconstants

By replacing equation (17) in (16) we have

_V minus h1x2

minus h2y2

minus h3z2

minus h4e2d

minus h5e2e minus h6e

2k minus h7e

2l minus h8e

2m minus h9e

2c

minus h10e2δ minus h11e

2ρ lt 0

(18)

which is the negative definite function on R11 for positiveconstants hi (i 1 11)

0ndash5ndash10 5 10x

4

3

2

1

0

ndash1

ndash2

y

(a)

0ndash5ndash10 5 10x

6

4

2

0

ndash2

ndash4

ndash6

z

(b)

500 600 700 800 900t

10

5

0

ndash5

ndash10

x

(c)

500 600 700 800 900t

4

3

2

1

0

ndash1

ndash2

y

(d)

Figure 1 Phase diagrams and time histories of system (6) (a) projected onto x-y phase plane (b) projected onto x-z phase plane (c) timehistory of (x) and (d) time history of y

4 Mathematical Problems in Engineering

So we have found a function which verifies the Lya-punov stability theorem (Vgt 0 _Vlt 0)

us we have the following result

Theorem 1 5e new financial system (8) with the unknownparameters is globally and asymptotically stabilized at theorigin for all initial conditions by the adaptive law (9) wherethe update law of the parameters is given by (17) with hi (i

1 11) being positive constants

312 Simulation Results In this part we use the fourth-order RungendashKutta algorithm to solve the new financialsystem (8) with the adaptive law (9) and the parameters

update law (17) For the simulation the time-step h 0001is chosen e initial conditions (x(0) y(0) z(0))

(12 15 16) are used e parameters of the new financialsystem are chosen as follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(19)

For hi (1 11) of the adaptive and update laws wechoose hi 3

For the initial value of the estimated parameters weassume the following values

e(0) 3 k(0) 1 1113954c(0) 4 m(0) 3 l(0) 1 ρ(0) 2 d(0) 1 δ(0) 3 (20)

By applying the adaptive control law (9) and the pa-rameter update law (17) to the new controlled financial system(8) the results of the numerical simulations are shown in

Figures 3 and 4 From Figure 3 it can be seen that the statevariables of the system converge asymptotically towards theorigin (zero) Figure 4 shows the estimated parameters for

4

3

2

1

0

ndash1

ndash20ndash5ndash10 5 10x

y

(a)

0ndash5ndash10 5 10x

z

4

2

0

ndash2

ndash4

(b)

200 300 400 500 600t

x

5

0

ndash5

ndash10

(c)

4

3

2

1

0

ndash2

ndash1

200 300 400 500 600t

y

(d)

Figure 2 Phase diagrams and time histories of system (7) with q1 1 q2 088 and q3 1 (a) projected onto x-y phase plane(b) projected onto x-z phase plane (c) time history of (x) and (d) time history of y

Mathematical Problems in Engineering 5

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

2

1

0

ndash1

2

1

0

ndash1

2

1

0

ndash1

x(t)

y(t)

z(t)

Figure 3 Time histories of the controlled integer order new financial system (8)

4

35

3

25

2

15

1

05

0

ndash05

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

δhat = 1 = δ

lhat = 01 = l

ehat = 03 = e

ρhat = 005 = ρ

Figure 4 Parameter estimates for adaptive control in integer order case

6 Mathematical Problems in Engineering

adaptive control in integer order case and as it can be seenthese parameters converge towards the real values of theparameters of the system ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(21)

32 Fractional Order Case In this part the adaptive controlof the new fractional order financial system is performed toglobally stabilize the new financial system with fractionalorder

321 Controller Design Consider the following new frac-tional order controlled financial system

Dq1x dz +(y minus e)x + u1

Dq2y minus ky

2minus lx

2+ m + u2

Dq3z minus cz minus δx minus ρy + u3

⎧⎪⎪⎨

⎪⎪⎩(22)

in which the ui (i 1 2 3) are adaptive controllers whichwill be subsequently designed taking into account the satevariables of the system and the estimation of the unknownconstant parameters d e k l m c δ and ρ of the system

To allow the states of the system to converge asymp-totically to the origin we take the following adaptive controlfunctions

u1 minus yx minus 1113954dz + 1113954ex minus h1Dq1minus 1

x + v1

u2 1113954ky2

+1113954lx2

minus 1113954m minus h2Dq2minus 1

y + v2

u3 1113954cz + 1113954δx + 1113954ρy minus h3Dq3minus 1

z + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(23)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimation of theunknown constant parametersd e k l m c δ and ρ re-spectively e hi (i 1 2 3) are positive constants andvi (i 1 2 3) are nonlinear functions that will be designed

By replacing the control law (23) in system (22) we have

Dq1x (d minus 1113954d)z minus (e minus 1113954e)x minus h1D

q1minus 1x + v1

Dq2y minus (k minus 1113954k)y

2minus (l minus 1113954l)x

2+(m minus 1113954m) minus h2D

q2minus 1y + v2

Dq3z minus (c minus 1113954c)z minus (δ minus 1113954δ)x minus (ρ minus 1113954ρ)y

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(24)

With the estimation error of unknown parameters de-fined by equation (11) we obtain

Dq1x edz minus eex minus h1D

q1minus 1x + v1

Dq2y minus eky

2minus elx

2+ em minus h2D

q2minus 1y + v2

Dq3z minus ecz minus eδx minus eρy minus h3D

q3minus 1z + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(25)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider Lyapunov quadratic function definedas follows

V 12

x2

+ y2

+ z2

+ e2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(26)

which is a positive definite function on R11e derivative with respect to time of equation (26) gives

us

_V x _x + y _y + z _z + ed _ed + ee _ee + ek _ek

+ el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

xD1minus q1D

q1x + yD1minus q2D

q2y + zD1minus q3D

q3z

+ ed _ed + ee _ee + ek _ek + el _el + em _em

+ ec _ec + eδ _eδ + eρ _eρ

(27)

Taking into account system (25) and equation (15)equation (27) becomes

_V xD1minus q1 edz minus eex minus h1D

q1minus 1x + v11113960 1113961

+ yD1minus q2 minus eky

2minus elx

2+ em minus h2D

q2minus 1y + v21113960 1113961

+ zD1minus q3 minus ecz minus eδx minus eρy minus h3D

q3minus 1z + v31113960 1113961

minus_1113954ded minus _1113954eee minus

_1113954kek minus_1113954lel minus _1113954mem minus _1113954cec

minus_1113954δeδ minus _1113954ρeρ

(28)

From equation (28) we deduce that the estimated pa-rameters update law is

_1113954d h4ed

_1113954e h5ee

_1113954k h6ek

_1113954l h7el

_1113954m h8em

_1113954c h9ec

_1113954δ h10eδ

_1113954ρ h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(29)

in which h4 h5 h6 h7 h8 h9 h10 and h11 are positiveconstants

From equation (28) we also deduce that the nonlinearfunctions vi (i 1 2 3) are given by

v1 minus edz + eex

v2 eky2

+ elx2

minus em

v3 ecz + eδx + eρy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)

By replacing equations (29) and (30) in equation (28) weget

Mathematical Problems in Engineering 7

_V minus h1x2

minus h2y2

minus h3z2

minus h4e2d

minus h5e2e minus h6e

2k minus h7e

2l minus h8e

2m minus h9e

2c

minus h10e2δ minus h11e

2ρ lt 0

(31)

which is a negative definite function on R11 for positiveconstants hi (i 1 11)

So we have found a function which verifies the Lya-punov stability theorem Vgt 0 _Vlt 0

us we have the following result

Theorem 2 5e new fractional order financial system (22)with the unknown parameters is globally and asymptoticallystabilized at the origin for all initial conditions by the adaptivelaw (23) with the vi (i 1 2 3) given by (30) and where theparameters update law is given by (29) with hi (i 1 11)

being positive constants

322 Simulation Results In this part we use the Adams-Bashforth-Moulton predictor-corrector method proposedby Diethelm et al [35] to solve the new fractional orderfinancial system (1922) with the adaptive control law (23)the vi (i 1 2 3) given by (30) and the parameter updatelaw given by (29) For the simulation the time-step h

0001 is chosen e initial conditions (x(0) y(0) z(0))

(12 15 16) are usede orders qi (i 1 2 3) are taken asfollows (q1 q2 q3) (1 088 1) ie the case of incom-mensurate order

e parameters of the new financial system are chosen asfollows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(32)

For hi (i 1 11) of the adaptive and update lawswe choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) 4 1113954k(0) 2 1113954c(0) 3 1113954m(0) 51113954l(0) 2 1113954ρ(0) 3 1113954d(0) 7 1113954δ(0) 8 (33)

By applying the adaptive law (23) and the parametersupdate law (29) to the new fractional order controlledfinancial system (22) the results of numerical simulationsare shown in Figures 5 and 6 From Figure 5 it can be seenthat the state variables of the system converge asymp-totically towards the origin Figure 6 shows the estimatedparameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ which as it can be seenconverge towards the real values of the parameters of thesystem ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(34)

4 Adaptive Synchronization of the IdenticalNew Financial System

41 Integer Order Case In this part the identical adaptivesynchronization of the new integer order financial system isachieved

411 Analytical Results Let us consider the master systemas being the system described with index 1 and the slavesystem as being the system described with index 2 We havetherefore for the master system the system

_x1 dz1 + y1 minus e( 1113857x1

_y1 minus ky21 minus lx

21 + m

_z1 minus cz1 minus δx1 minus ρy1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(35)

and for slave system we have

_x2 dz2 + y2 minus e( 1113857x2 + u1

_y2 minus ky22 minus lx

22 + m + u2

_z2 minus cz2 minus δx2 minus ρy2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(36)

in which u1 u2 u3 are controllers to be designed so thatsystem (36) synchronizes with system (35) andd e k l m c δ and ρ are the unknown constant parametersof the system

Let us define the error functions between the statevariables of systems (36) and (35) as follows

e1 x2 minus x1

e2 y2 minus y1

e3 z2 minus z1

(37)

From equation (37) we obtain the following errorsystem

_e1 minus ee1 + de3 + y2x2 minus y1x1 + u1

_e2 minus k y22 minus y

211113872 1113873 minus l x

22 minus x

211113872 1113873 + u2

_e3 minus ce3 minus δe1 minus ρe2 + u3

⎧⎪⎪⎨

⎪⎪⎩(38)

Let us define the adaptive control functionsui (i 1 2 3) as follows

u1 1113954ee1 minus 1113954de3 minus y2x2 + y1x1 minus h1e1

u2 1113954k y22 minus y

211113872 1113873 +1113954l x

22 minus x

211113872 1113873 minus h2e2

u3 1113954ce3 + 1113954δe1 + 1113954ρe2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(39)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimates of theparameters d e k l m c δ and ρ respectively and hi (i

1 2 3) are positive constants

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

2

1

0

ndash1

2

1

0

ndash1

2

1

0

ndash1

x(t)

y(t)

z(t)

Figure 5 Time histories of the controlled fractional order new financial system (22)

7

8

6

5

4

3

2

1

0

ndash10 1 2 3 4 5 6 7 8 9 10

Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

δhat = 1 = δlhat = 01 = l ehat = 03 = e

ρhat = 005 = ρ

Para

met

er es

timat

es

Figure 6 Parameter estimates for adaptive control in fractional order case

Mathematical Problems in Engineering 9

By replacing the control law (39) in (38) we have

_e1 minus (e minus 1113954e)e1 +(d minus 1113954d)e3 minus h1e1

_e2 minus (k minus 1113954k) y22 minus y

211113872 1113873 minus (l minus 1113954l) x

22 minus x

211113872 1113873 minus h2e2

_e3 minus (c minus 1113954c)e3 minus (δ minus 1113954δ)e1 minus (ρ minus 1113954ρ)e2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(40)

Define the estimation error of unknown parameters asfollows

ed d minus 1113954d

ee e minus 1113954e

ek k minus 1113954k

el l minus 1113954l

em m minus 1113954m

ec c minus 1113954c

eδ δ minus 1113954δ

eρ ρ minus 1113954ρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(41)

By substituting equation (41) in system (40) system (40)becomes

_e1 minus eee1 + ede3 minus h1e1

_e2 minus ek y22 minus y

211113872 1113873 minus el x

22 minus x

211113872 1113873 minus h2e2

_e3 minus ece3 minus eδe1 minus eρe2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(42)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider the quadratic Lyapunov functiondefined as follows

V 12

e21 + e

22 + e

23 + e

2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(43)

which is a positive definite function on R11

e derivative with respect to time of equation (43) givesus

_V e1 _e1 + e2 _e2 + e3 _e3 + ed _ed + ee _ee

+ ek _ek + el _el + em _em + ec _ec + eδ _eδ + eρ _eρ(44)

Using system (42) and equation (15) equation (44)becomes

_V minus h1e21 minus h2e

22 minus h3e

23 + ee minus e

21 minus _1113954e1113872 1113873

+ed e1e3 minus_1113954d1113874 1113875 + ek minus e2 y

22 minus y

211113872 1113873 minus

_1113954k1113876 1113877

+el minus e2 x22 minus x

211113872 1113873 minus

_1113954l1113876 1113877 + ec minus e23 minus _1113954c1113872 1113873

+eδ minus e3e1 minus_1113954δ1113874 1113875 + eρ minus e3e2 minus _1113954ρ1113872 1113873 minus _1113954mem

(45)

From equation (45) we deduce that the estimated pa-rameters update law is

_1113954d e1e3 + h4ed_1113954e minus e

21 + h5ee

_1113954k minus e2 y22 minus y

211113872 1113873 + h6ek

_1113954l minus e2 x22 minus x

211113872 1113873 + h7el

_1113954m h8em_1113954c minus e

23 + h9ec

_1113954δ minus e3e1 + h10eδ_1113954ρ minus e3e2 + h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(46)

in which h4 h5 h6 h7 h8 h9 h10 and h11 are positiveconstants

By replacing equation (46) in equation(45) we get

_V minus h1e21 minus h2e

22 minus h3e

23 minus h4e

2d minus h5e

2e

minus h6e2k minus h7e

2l minus h8e

2m minus h9e

2c minus h10e

2δ minus h11e

2ρ lt 0

(47)

which is a negative definite function on R11 for positiveconstants hi (i 1 11) So we have found a functionwhich verifies the Lyapunov stability theorem (Vgt 0 _Vlt 0)

us the error functions converge globally and as-ymptotically towards zero for all initial conditions esynchronization of the states of the identical systems (35)and (36) is therefore complete So we have the followingresult

Theorem 3 5e identical financial systems (35) and (36)with unknown parameters are globally and asymptoticallysynchronized for all initial conditions by the adaptive controllaw (39) where the parameters update law is given by (46) andthe hi (i 1 11) are positive constants

412 Simulation Results In this part we use the fourth-order Rungendash-Kutta algorithm to solve the two identicalfinancial systems (35) and (36) with the adaptive control law(39) and the parameters update law (46) For the simulationthe time-step h 0001 is chosen e initial conditions forthe master system are (x1(0) y1(0) z1(0)) (12 15 16)

and for the slave system (x2(0) y2(0) z2(0))

(02 05 06)e parameters of the new financial system are chosen as

follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(48)

For the hi (i 1 11) of the adaptive and updatelaws we choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) minus 1 1113954k(0) 2 1113954c(0) 05

1113954m(0) 41113954l(0) 2 1113954ρ(0) 3

1113954d(0) 7 1113954δ(0) minus 05

(49)

10 Mathematical Problems in Engineering

8

4

0

ndash4

x 1(t)

x2(t)

x1(t)

x2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(a)

0 1 2 3 4 5 6 7 8 9 10Time t

3

2

1

0

y 1(t)

y2(t)

y1(t)

y2(t)

(b)

0 1 2 3 4 5 6 7 8 9 10Time t

24

0ndash2ndash4

z 1(t)

z2(t)

z1(t)

z2(t)

(c)

0 1 2 3 4 5 6 7 8 9 10Time t

1

0

ndash1

ndash2e 1(t)

e2(t)

e 3(t)

e1(t)

e2(t)e3(t)

(d)

Figure 7 (andashc) Time evolutions of the master and slave systems state variables (x1 x2) (y1 y2) (z1 z2) respectively in integer order caseand (d) time evolution of the error functions e1(black line) e2(red line) and e3(blue line)

7

6

5

4

3

2

1

0

ndash1

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

ρhat = 005 = ρ

δhat = 1 = δlhat = 01 = l ehat = 03 = e

khat = 002 = k

Figure 8 Parameter estimates for adaptive synchronization in integer order case

Mathematical Problems in Engineering 11

By applying the adaptive control law (39) and the pa-rameter update law (46) to the new controlled financialsystem (36) the results of the numerical simulations areshown in Figures 7(a)ndash7(d) and 8 From Figures 7(a)ndash7(c) itcan be seen that the state variables of the slave and mastersystems are synchronized Figure 7(d) shows the errorsystem which eventually converges to zero Finally Figure 8shows the estimated parameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρwhich as it can be seen converge towards the real values ofthe parameters of the system ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(50)

42FractionalOrderCase In this part the identical adaptivesynchronization of the new fractional order financial systemis achieved

421 Analytical Results Let us consider the master systemas being the system described with index 1 and the slavesystem as being the system described with index 2 We havetherefore for the master system the system

Dq1x1 dz1 + y1 minus e( 1113857x1

Dq2y1 minus ky

21 minus lx

21 + m

Dq3z1 minus cz1 minus δx1 minus ρy1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(51)

and for slave system we have

Dq1x2 dz2 + y2 minus e( 1113857x2 + u1

Dq2y2 minus ky

22 minus lx

22 + m + u2

Dq3z2 minus cz2 minus δx2 minus ρy2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(52)

where u1 u2 u3 are controllers to be designed so that system(52) synchronizes with system (51) and d e k l m

c δ and ρ are the unknown constant parameters of thesystem

Let us define the error functions between the statevariables of systems (2) and (51) as follows

e1 x2 minus x1

e2 y2 minus y1

e3 z2 minus z1

(53)

From equation (53) we get the following error system

Dq1e1 minus ee1 + de3 + y2x2 minus y1x1 + u1

Dq2e2 minus k y

22 minus y

211113872 1113873 minus l x

22 minus x

211113872 1113873 + u2

Dq3e3 minus ce3 minus δe1 minus ρe2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(54)

Let us define the adaptive control functionsui (i 1 2 3)

u1 1113954ee1 minus 1113954de3 minus y2x2 + y1x1 minus h1Dq1minus 1

e1 + v1

u1 1113954k y22 minus y

211113872 1113873 +1113954l x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

u3 1113954ce3 + 1113954δe1 + 1113954ρe2 minus h3Dq3minus 1

e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(55)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimates of theparameters d e k l m c δ and ρ respectively hi (i

1 2 3) are positive constants and vi (i 1 2 3) are non-linear functions that will be designed By replacing thecontrol law (55) in (54) we get

Dq1e1 minus (e minus 1113954e)e1 +(d minus 1113954d)e3 minus h1D

q1minus 1e1 + v1

Dq2e2 minus (k minus 1113954k) y

22 minus y

211113872 1113873 minus (l minus 1113954l) x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

Dq3e3 minus (c minus 1113954c)e3 minus (δ minus 1113954δ)e1 minus (ρ minus 1113954ρ)e2 minus h3D

q3minus 1e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(56)

With the estimation error of unknown parameters de-fined by equation (41) we obtain

Dq1e1 minus eee1 + ede3 minus h1D

q1minus 1e1 + v1

Dq2e2 minus ek y

22 minus y

211113872 1113873 minus el x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

Dq3e3 minus ece3 minus eδe1 minus eρe2 minus h3D

q3minus 1e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(57)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider the quadratic Lyapunov functiondefined as follows

V 12

e21 + e

22 + e

23 + e

2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(58)

which is a positive definite function on R11 e derivativewith respect to time of equation (58) gives us

_V e1 _e1 + e2 _e2 + e3 _e3 + ed _ed + ee _ee

+ ek _ek + el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

e1D1minus q1D

q1e1 + e2D1minus q2D

q2e2

+ e3D1minus q3D

q3e3 + ed _ed + ee _ee + ek _ek

+ el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

(59)

Taking into account system (57) and equation (15)equation (59) becomes

_V e1D1minus q1 minus eee1 + ede3 minus h1D

q1minus 1e1 + v11113960 1113961

+ e2D1minus q2 minus ek y

22 minus y

211113872 1113873 minus el x

22 minus x

211113872 11138731113960

minus h2Dq2minus 1

e2 + v21113961 + e3D1minus q3 minus ece3 minus eδe11113960

minus eρe2 minus h3Dq3minus 1

e3 + v31113961 minus_ded minus _eee minus

_kek

minus_lel minus _mem minus _1113954cec minus

_δeδ minus _ρeρ

(60)

From equation (60) we deduce that the estimated pa-rameters update law is

12 Mathematical Problems in Engineering

_1113954d h4ed

_1113954e h5ee

_1113954k h6ek

_1113954l h7el

_1113954m h8em

_1113954c h9ec

_1113954δ h10eδ

_1113954ρ h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

where h4 h5 h6 h7 h8 h9 h10 and h11 are positive constantsFrom equation (60) we also deduce that the nonlinear

functions vi (i 1 2 3) are given by

v1 eee1 minus ede3

v2 ek y22 minus y

211113872 1113873 + el x

22 minus x

211113872 1113873

v3 ece3 + eδe1 + eρe2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(62)

By replacing equations (61) and (62) in (60) we get

_V minus h1e21 minus h2e

22 minus h3e

23 minus h4e

2d minus h5e

2e

minus h6e2k minus h7e

2l minus h8e

2m minus h9e

2c minus h10e

2δ

minus h11e2ρ lt 0

(63)

which is a negative definite function on R11 for positiveconstants hi (i 1 11) So we have found a functionwhich verifies the Lyapunov stability theorem (Vgt 0 _Vlt 0)

us the error functions converge globally and as-ymptotically towards zero for all initial conditions esynchronization of the states of the identical systems (51)and (52) is therefore complete So we have the followingresult

Theorem 4 5e identical financial systems (51) and (52)with unknown parameters are globally and asymptoticallysynchronized for all initial conditions by the adaptive controllaw (55) with vi (i 1 2 3) given by (62) and where theparameters update law is given by (61) and thehi (i 1 11) are positive constants

422 Simulation Results In this part we use the Adams-Bashforth-Moulton predictor-corrector method proposedby Diethelm et al [35] to solve the two identical fractionalorder systems (51) and (52) with the adaptive control law(55) the vi (i 1 2 3) given by (62) and the law forupdating the parameters is given by (61) For the simulationthe time-step h 0001 is chosen e initial conditions forthe master system are (x1(0) y1(0) z1(0)) (12 15 16)

and for the slave system (x2(0) y2(0) z2(0))

(02 05 06) are usede orders qi (i 1 2 3) are taken asfollows (q1 q2 q3) (1 088 1) ie the case of incom-mensurate order

e parameters of the new fractional order financialsystem are chosen as follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(64)

0 1 2 3 4 5 6 7 8 9 10

8

4

0

ndash4

x 1(t)

x2(t)

x1(t)

x2(t)

Time t

(a)

3

2

1

0

y 1(t)

y2(t)

y1(t)

y2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(b)

20

ndash2ndash4

z 1(t)

z2(t)

z1(t)

z2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(c)

2

1

0

ndash1e 1(t)

e2(t)

e 3(t)

e1(t)

e2(t)e3(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(d)

Figure 9 (andashc) Time evolutions of the master and slave systems state variables (x1 x2) (y1 y2) (z1 z2) respectively in fractional ordercase and (d) time evolution of the error functions e1(black line) e2(red line) and e3(blue line)

Mathematical Problems in Engineering 13

For hi (i 1 11) of the adaptive and update lawswe choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) 1 1113954k(0) minus 1 1113954c(0) 5 1113954m(0) 61113954l(0) 3 1113954ρ(0) 4 1113954d(0) 3 1113954δ(0) 7 (65)

By applying the adaptive control law (55) and the parameterupdate law (61) to the new controlled fractional-order financialsystem (52) the results of numerical simulations are shown inFigures 9(a)ndash9(d) and 10 From Figures 9(a)ndash9(c) it can be seenthat the state variables of the master and slave systems aresynchronized Figure 9(d) shows the error system whicheventually converges to zero Finally Figure 10 shows the es-timated parameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ which as it can beseen converge towards the real values of the parameters of thesystem ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(66)

5 Conclusions

In this paper the adaptive control and the adaptive syn-chronization of a new financial system with unknownconstant parameters were studied in the cases of integer andfractional order e adaptive control law and the adaptivesynchronization law were designed based on Lyapunovrsquosstability theory and on the adaptive control theory e lawshave been designed in the cases of integer and incom-mensurate fractional order system

e proposed adaptive control technique is effective forchaos control and synchronization of the new financial

system when the constant parameters of the system areunknown Numerical simulations are carried out to provethe efficiency of the control and synchronization techniquesdesigned in this work

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank IMSP-UAC and the German AcademicExchange Service (DAAD) for financial support under theprogramme ldquoIn-CountryIn-Region Scholarship Programmerdquo

References

[1] K B Oldham and J Spanier 5e Fractional Calculus Aca-demic Press New York NY USA 1974

[2] B Ross ldquoFractional calculus and its applicationsrdquo LectureNotes in Mathematics Vol 457 Springer-Verlag New YorkNY USA 1975

7

6

5

4

3

2

1

0

ndash1

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

ρhat = 005 = ρ δhat = 1 = δlhat = 1 = l ehat = 03 = e

Figure 10 Parameter estimates for adaptive synchronization in fractional order case

14 Mathematical Problems in Engineering

[3] V Daftardar-Gejji and S Bhalekar ldquoChaos in fractional or-dered Liu systemrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1117ndash1127 2010

[4] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons amp Fractals vol 36no 5 pp 1305ndash1314 2008

[5] K Rajagopal A Bayani S Jafari A Karthikeyan andI Hussain ldquoChaotic dynamics of a fractional order glucose-insulin regulatory systemrdquo Frontiers of Information Tech-nology amp Electronic Engineering vol 21 no 7 pp 1108ndash11182019

[6] T T Hartley C F Lorenzo and H Killory Qammer ldquoChaosin a fractional order Chuarsquos systemrdquo IEEE Transactions onCircuits and Systems I Fundamental 5eory and Applicationsvol 42 no 8 pp 485ndash490 1995

[7] Q Jia ldquoChaos control and synchronization of the Newton-Leipnik chaotic systemrdquo Chaos Solitons amp Fractals vol 35no 4 pp 814ndash824 2008

[8] M T Yassen ldquoAdaptive control and synchronization of amodified Chuarsquos circuit systemrdquo Applied Mathematics andComputation vol 135 no 1 pp 113ndash128 2003

[9] V Sundarapandian ldquoAdaptive control and synchronizationof a generalized Lotka-Volterra systemrdquo International Journalof Bioinformatics amp Biosciences vol 1 no 1 pp 1ndash12 2011

[10] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A StatisticalMechanics and Its Applications vol 389 no 12 pp 2434ndash2442 2010

[11] U E Kocamaz A Goksu H Taskin and Y UyarogluldquoControl of chaotic two-predator one-prey model with singlestate control signalsrdquo Journal of Intelligent Manufacturingpp 1ndash10 2020

[12] A Hajipour and H Tavakoli ldquoDynamic analysis and adaptivesliding mode controller for a chaotic fractional incommen-surate order financial systemrdquo International Journal of Bi-furcation and Chaos vol 27 no 13 p 14 2017

[13] X Yi R Guo and Y Qi ldquoStabilization of chaotic systems withboth uncertainty and disturbance by the UDE-based controlmethodrdquo IEEE Access vol 8 no 1 pp 62471ndash62477 2020

[14] L Liu B Li and R Guo ldquoConsensus control for networkedmanipulators with switched parameters and topologiesrdquo IEEEAccess vol 9 pp 9209ndash9217 2021

[15] T Hou Y Liu and F Deng ldquoFinite horizon H2∕Hinfin controlfor SDEs with infinite Markovian jumpsrdquo Nonlinear AnalysisHybrid Systems vol 34 pp 108ndash120 2019

[16] R Xu and F Zhang ldquo-Nash mean-field games for generallinear-quadratic systems with applicationsrdquo Automaticavol 114 pp 1ndash4 2020

[17] R HilferApplications of Fractional Calculus in Physics WorldScientific Hackensack NJ USA 2001

[18] R He and P G Vaidya ldquoImplementation of chaotic cryp-tography with chaotic synchronizationrdquo Physical Review Evol 57 no 2 pp 1532ndash1535 1998

[19] J H Park ldquoChaos synchronization of a chaotic system vianonlinear controlrdquo Chaos Solitons amp Fractals vol 25 no 3pp 579ndash584 2005

[20] L Huang R Feng andMWang ldquoSynchronization of chaoticsystems via nonlinear controlrdquo Physics Letters A vol 320no 4 pp 271ndash275 2004

[21] S Bhalekar and V Daftardar-Gejji ldquoSynchronization ofdifferent fractional order chaotic systems using active con-trolrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3536ndash3546 2010

[22] H N Agiza and M T Yassen ldquoSynchronization of Rosslerand Chen chaotic dynamical systems using active controlrdquoPhysics Letters A vol 278 no 4 pp 191ndash197 2001

[23] T-l Liao ldquoAdaptive synchronization of two Lorenz SystemsrdquoChaos Solitons amp Fractals vol 9 no 9 pp 1555ndash1561 1998

[24] H Yu G Cai and Y Li ldquoDynamic analysis and control of anew hyperchaotic finance systemrdquo Nonlinear Dynamicsvol 67 no 3 pp 2171ndash2182 2012

[25] J-h Ma and Y-s Chen ldquoStudy for the bifurcation topologicalstructure and the global complicated character of a kind ofnonlinear finance system Irdquo Applied Mathematics and Me-chanics vol 22 no 11 pp 1240ndash1251 2001

[26] M S Abd-Elouahab N E Hamri and J Wang ldquoChaoscontrol of a fractional order financial systemrdquo MathematicalProblems in Engineering vol 2010 Article ID 27064618 pages 2010

[27] I Hernandez C Mateos J Nuntildeez and A F Tenorio ldquoLietheory applications to problems in mathematical finance andeconomicsrdquo Applied Mathematics and Computation vol 208no 2 pp 446ndash452 2009

[28] S A David J A T Machado D D Quintino andJ M Balthazar ldquoPartial chaos suppression in a fractionalorder macroeconomic modelrdquo Mathematics and Computersin Simulation vol 122 pp 55ndash68 2016

[29] J Yang E Zhang and M Liu ldquoBifurcation analysis and chaoscontrol in a modified finance system with delayed feedbackrdquoInternational Journal of Bifurcation and Chaos vol 26 no 062016

[30] A Hajipour and H Tavakoli ldquoAnalysis and circuit simulationof a novel nonlinear fractional incommensurate order fi-nancial systemrdquo Optik vol 127 no 22 pp 10643ndash106522016

[31] X Zhao Z Li and S Li ldquoSynchronization of a chaotic financesystemrdquo Applied Mathematics and Computation vol 217no 13 pp 6031ndash6039 2011

[32] Y Liao Y Zhou F Xu and X-B Shu ldquoA study on thecomplexity of a new chaotic financial systemrdquo Complexityvol 2020 Article ID 8821156 5 pages 2020

[33] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

[34] I Petras Fractional-order Nonlinear Systems ModelingAnalysis and Simulation Springer Science amp Business MediaBerlin Germany 2011

[35] K Diethelm N J Ford andA D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquo Nonlinear Dynamics vol 29 no 1ndash4 pp 3ndash222002

Mathematical Problems in Engineering 15

dx

dt dz +(y minus e)x

dy

dt minus ky

2minus lx

2+ m

dz

dt minus cz minus δx minus ρy

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

where the parameters e k c m l ρ and δ are constantsIn [32] when e 03 k 002 c 1 m 1 l 01 ρ

005 andd 12 δ 1 and initial conditions (12 15 16)are considered system (6) exhibits a chaotic behavior asshown in Figures 1(a)ndash1(d)When the initial conditions (0205 and 06) are considered system (6) also presents achaotic behavior [32]

e generalization of system (6) ie the fractional orderversion of the new financial system is also considered in thisstudy Classical derivatives (integer order) are replaced byfractional order derivatives as follows

Dq1x dz +(y minus e)x

Dq2y minus ky

2minus lx

2+ m

Dq3z minus cz minus δx minus ρy

⎧⎪⎪⎨

⎪⎪⎩(7)

where qi isin (0 1) and Dqi (dqi dtqi )(i 1 2 3) Ifq1 q2 q3 q then system (7) is said to be a commen-surate order system otherwise it is said to be an incom-mensurate order system

e new fractional order financial system is chaotic whenthe values of the above parameters are considered the initialconditions (12 15 16) and the orders q1 1 q2 088

and q3 1 are considerede phase diagrams projected onto the phase planes

(x y) (x z) and the time histories of the state variablesx(t) and y(t) are shown in Figures 2(a)ndash2(d)

3 Adaptive Control of Chaos in a NewFinancial System

31 Integer Order Case In this part we will be interested inthe design of an adaptive control law in order to globallystabilize the new integer order financial system

311 Controller Design To control the chaos in system (6)adaptive controllers are added to it e new controlledfinancial system can therefore be written in the followingform

_x dz +(y minus e)x + u1

_y minus ky2

minus lx2

+ m + u2

_z minus cz minus δx minus ρy + u3

⎧⎪⎪⎨

⎪⎪⎩(8)

in which _x (dxdt) _y (dydt) and _z (dzdt) eui (i 1 2 3) are adaptive controllers which will be sub-sequently designed taking into account the state variables ofthe system and the estimation of the unknown constantparameters d e k l m c δ and ρ of the system

To allow the state variables of the system to convergeasymptotically to the origin we take the following adaptivecontrol functions

u1 minus yx minus 1113954dz + 1113954ex minus h1x

u2 1113954ky2

+ 1113954lx2

minus 1113954m minus h2y

u3 1113954cz + 1113954δx + 1113954ρy minus h3z

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(9)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimation of theunknown constant parameters d e k l m c δ and ρ re-spectively and hi (i 1 2 3) are positive constants

By replacing the control law (9) in system (8) we have

_x (d minus 1113954d)z minus (e minus 1113954e)x minus h1x

_y minus (k minus 1113954k)y2

minus (l minus 1113954l)x2

+(m minus 1113954m) minus h2y

_z minus (c minus 1113954c)z minus (δ minus 1113954δ)x minus (ρ minus 1113954ρ)y minus h3z

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(10)

Let us define the estimation error of unknown param-eters as follows

ed d minus 1113954d

ee e minus 1113954e

ek k minus 1113954k

el l minus 1113954l

em m minus 1113954m

ec c minus 1113954c

eδ δ minus 1113954δ

eρ ρ minus 1113954ρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(11)

By replacing equation (11) in system (10) we have

_x edz minus eex minus h1x

_y minus eky2

minus elx2

+ em minus h2y

_z minus ecz minus eδx minus eρy minus h3z

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(12)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider the Lyapunov quadratic functiondefined as follows

V 12

x2

+ y2

+ z2

+ e2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(13)

which is a positive definite function on R11e derivative with respect to time of equation (13) gives

us_V x _x + y _y + z _z + ed _ed + ee _ee + ek _ek

+ el _el + em _em + ec _ec + eδ _eδ + eρ _eρ(14)

which specify that

Mathematical Problems in Engineering 3

_ed minus_1113954d

_ee minus _1113954e

_ek minus_1113954k _el minus

_1113954l _em minus _1113954m _ec minus _1113954c _eδ minus_1113954δ _eρ minus _1113954ρ

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(15)

Using system (12) and equation (15) equations (14)becomes

_V minus h1x2

minus h2y2

minus h3z2

+ ed(xz minus_1113954d)

+ ee minus x2

minus _1113954e1113872 1113873 + ek minus y3

minus_1113954k1113874 1113875

+ el minus yx2

minus_1113954l1113874 1113875 + em(y minus _1113954m)

+ ec minus z2

minus _1113954c1113872 1113873 + eδ(minus xz minus_1113954δ)

+ eρ(minus zy minus _1113954ρ)

(16)

From equation (16) we deduce that the estimated pa-rameters update law is

_d xz + h4ed

_e minus x2

+ h5ee

_k minus y

3+ h6ek

_l minus yx

2+ h7el

_m y + h8em

_c minus z2

+ h9ec

_δ minus xz + h10eδ_ρ minus zy + h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

in which h4 h5 h6 h7 h8 h9 h10 and h11 are positiveconstants

By replacing equation (17) in (16) we have

_V minus h1x2

minus h2y2

minus h3z2

minus h4e2d

minus h5e2e minus h6e

2k minus h7e

2l minus h8e

2m minus h9e

2c

minus h10e2δ minus h11e

2ρ lt 0

(18)

which is the negative definite function on R11 for positiveconstants hi (i 1 11)

0ndash5ndash10 5 10x

4

3

2

1

0

ndash1

ndash2

y

(a)

0ndash5ndash10 5 10x

6

4

2

0

ndash2

ndash4

ndash6

z

(b)

500 600 700 800 900t

10

5

0

ndash5

ndash10

x

(c)

500 600 700 800 900t

4

3

2

1

0

ndash1

ndash2

y

(d)

Figure 1 Phase diagrams and time histories of system (6) (a) projected onto x-y phase plane (b) projected onto x-z phase plane (c) timehistory of (x) and (d) time history of y

4 Mathematical Problems in Engineering

So we have found a function which verifies the Lya-punov stability theorem (Vgt 0 _Vlt 0)

us we have the following result

Theorem 1 5e new financial system (8) with the unknownparameters is globally and asymptotically stabilized at theorigin for all initial conditions by the adaptive law (9) wherethe update law of the parameters is given by (17) with hi (i

1 11) being positive constants

312 Simulation Results In this part we use the fourth-order RungendashKutta algorithm to solve the new financialsystem (8) with the adaptive law (9) and the parameters

update law (17) For the simulation the time-step h 0001is chosen e initial conditions (x(0) y(0) z(0))

(12 15 16) are used e parameters of the new financialsystem are chosen as follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(19)

For hi (1 11) of the adaptive and update laws wechoose hi 3

For the initial value of the estimated parameters weassume the following values

e(0) 3 k(0) 1 1113954c(0) 4 m(0) 3 l(0) 1 ρ(0) 2 d(0) 1 δ(0) 3 (20)

By applying the adaptive control law (9) and the pa-rameter update law (17) to the new controlled financial system(8) the results of the numerical simulations are shown in

Figures 3 and 4 From Figure 3 it can be seen that the statevariables of the system converge asymptotically towards theorigin (zero) Figure 4 shows the estimated parameters for

4

3

2

1

0

ndash1

ndash20ndash5ndash10 5 10x

y

(a)

0ndash5ndash10 5 10x

z

4

2

0

ndash2

ndash4

(b)

200 300 400 500 600t

x

5

0

ndash5

ndash10

(c)

4

3

2

1

0

ndash2

ndash1

200 300 400 500 600t

y

(d)

Figure 2 Phase diagrams and time histories of system (7) with q1 1 q2 088 and q3 1 (a) projected onto x-y phase plane(b) projected onto x-z phase plane (c) time history of (x) and (d) time history of y

Mathematical Problems in Engineering 5

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

2

1

0

ndash1

2

1

0

ndash1

2

1

0

ndash1

x(t)

y(t)

z(t)

Figure 3 Time histories of the controlled integer order new financial system (8)

4

35

3

25

2

15

1

05

0

ndash05

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

δhat = 1 = δ

lhat = 01 = l

ehat = 03 = e

ρhat = 005 = ρ

Figure 4 Parameter estimates for adaptive control in integer order case

6 Mathematical Problems in Engineering

adaptive control in integer order case and as it can be seenthese parameters converge towards the real values of theparameters of the system ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(21)

32 Fractional Order Case In this part the adaptive controlof the new fractional order financial system is performed toglobally stabilize the new financial system with fractionalorder

321 Controller Design Consider the following new frac-tional order controlled financial system

Dq1x dz +(y minus e)x + u1

Dq2y minus ky

2minus lx

2+ m + u2

Dq3z minus cz minus δx minus ρy + u3

⎧⎪⎪⎨

⎪⎪⎩(22)

in which the ui (i 1 2 3) are adaptive controllers whichwill be subsequently designed taking into account the satevariables of the system and the estimation of the unknownconstant parameters d e k l m c δ and ρ of the system

To allow the states of the system to converge asymp-totically to the origin we take the following adaptive controlfunctions

u1 minus yx minus 1113954dz + 1113954ex minus h1Dq1minus 1

x + v1

u2 1113954ky2

+1113954lx2

minus 1113954m minus h2Dq2minus 1

y + v2

u3 1113954cz + 1113954δx + 1113954ρy minus h3Dq3minus 1

z + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(23)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimation of theunknown constant parametersd e k l m c δ and ρ re-spectively e hi (i 1 2 3) are positive constants andvi (i 1 2 3) are nonlinear functions that will be designed

By replacing the control law (23) in system (22) we have

Dq1x (d minus 1113954d)z minus (e minus 1113954e)x minus h1D

q1minus 1x + v1

Dq2y minus (k minus 1113954k)y

2minus (l minus 1113954l)x

2+(m minus 1113954m) minus h2D

q2minus 1y + v2

Dq3z minus (c minus 1113954c)z minus (δ minus 1113954δ)x minus (ρ minus 1113954ρ)y

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(24)

With the estimation error of unknown parameters de-fined by equation (11) we obtain

Dq1x edz minus eex minus h1D

q1minus 1x + v1

Dq2y minus eky

2minus elx

2+ em minus h2D

q2minus 1y + v2

Dq3z minus ecz minus eδx minus eρy minus h3D

q3minus 1z + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(25)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider Lyapunov quadratic function definedas follows

V 12

x2

+ y2

+ z2

+ e2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(26)

which is a positive definite function on R11e derivative with respect to time of equation (26) gives

us

_V x _x + y _y + z _z + ed _ed + ee _ee + ek _ek

+ el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

xD1minus q1D

q1x + yD1minus q2D

q2y + zD1minus q3D

q3z

+ ed _ed + ee _ee + ek _ek + el _el + em _em

+ ec _ec + eδ _eδ + eρ _eρ

(27)

Taking into account system (25) and equation (15)equation (27) becomes

_V xD1minus q1 edz minus eex minus h1D

q1minus 1x + v11113960 1113961

+ yD1minus q2 minus eky

2minus elx

2+ em minus h2D

q2minus 1y + v21113960 1113961

+ zD1minus q3 minus ecz minus eδx minus eρy minus h3D

q3minus 1z + v31113960 1113961

minus_1113954ded minus _1113954eee minus

_1113954kek minus_1113954lel minus _1113954mem minus _1113954cec

minus_1113954δeδ minus _1113954ρeρ

(28)

From equation (28) we deduce that the estimated pa-rameters update law is

_1113954d h4ed

_1113954e h5ee

_1113954k h6ek

_1113954l h7el

_1113954m h8em

_1113954c h9ec

_1113954δ h10eδ

_1113954ρ h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(29)

in which h4 h5 h6 h7 h8 h9 h10 and h11 are positiveconstants

From equation (28) we also deduce that the nonlinearfunctions vi (i 1 2 3) are given by

v1 minus edz + eex

v2 eky2

+ elx2

minus em

v3 ecz + eδx + eρy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)

By replacing equations (29) and (30) in equation (28) weget

Mathematical Problems in Engineering 7

_V minus h1x2

minus h2y2

minus h3z2

minus h4e2d

minus h5e2e minus h6e

2k minus h7e

2l minus h8e

2m minus h9e

2c

minus h10e2δ minus h11e

2ρ lt 0

(31)

which is a negative definite function on R11 for positiveconstants hi (i 1 11)

So we have found a function which verifies the Lya-punov stability theorem Vgt 0 _Vlt 0

us we have the following result

Theorem 2 5e new fractional order financial system (22)with the unknown parameters is globally and asymptoticallystabilized at the origin for all initial conditions by the adaptivelaw (23) with the vi (i 1 2 3) given by (30) and where theparameters update law is given by (29) with hi (i 1 11)

being positive constants

322 Simulation Results In this part we use the Adams-Bashforth-Moulton predictor-corrector method proposedby Diethelm et al [35] to solve the new fractional orderfinancial system (1922) with the adaptive control law (23)the vi (i 1 2 3) given by (30) and the parameter updatelaw given by (29) For the simulation the time-step h

0001 is chosen e initial conditions (x(0) y(0) z(0))

(12 15 16) are usede orders qi (i 1 2 3) are taken asfollows (q1 q2 q3) (1 088 1) ie the case of incom-mensurate order

e parameters of the new financial system are chosen asfollows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(32)

For hi (i 1 11) of the adaptive and update lawswe choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) 4 1113954k(0) 2 1113954c(0) 3 1113954m(0) 51113954l(0) 2 1113954ρ(0) 3 1113954d(0) 7 1113954δ(0) 8 (33)

By applying the adaptive law (23) and the parametersupdate law (29) to the new fractional order controlledfinancial system (22) the results of numerical simulationsare shown in Figures 5 and 6 From Figure 5 it can be seenthat the state variables of the system converge asymp-totically towards the origin Figure 6 shows the estimatedparameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ which as it can be seenconverge towards the real values of the parameters of thesystem ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(34)

4 Adaptive Synchronization of the IdenticalNew Financial System

41 Integer Order Case In this part the identical adaptivesynchronization of the new integer order financial system isachieved

411 Analytical Results Let us consider the master systemas being the system described with index 1 and the slavesystem as being the system described with index 2 We havetherefore for the master system the system

_x1 dz1 + y1 minus e( 1113857x1

_y1 minus ky21 minus lx

21 + m

_z1 minus cz1 minus δx1 minus ρy1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(35)

and for slave system we have

_x2 dz2 + y2 minus e( 1113857x2 + u1

_y2 minus ky22 minus lx

22 + m + u2

_z2 minus cz2 minus δx2 minus ρy2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(36)

in which u1 u2 u3 are controllers to be designed so thatsystem (36) synchronizes with system (35) andd e k l m c δ and ρ are the unknown constant parametersof the system

Let us define the error functions between the statevariables of systems (36) and (35) as follows

e1 x2 minus x1

e2 y2 minus y1

e3 z2 minus z1

(37)

From equation (37) we obtain the following errorsystem

_e1 minus ee1 + de3 + y2x2 minus y1x1 + u1

_e2 minus k y22 minus y

211113872 1113873 minus l x

22 minus x

211113872 1113873 + u2

_e3 minus ce3 minus δe1 minus ρe2 + u3

⎧⎪⎪⎨

⎪⎪⎩(38)

Let us define the adaptive control functionsui (i 1 2 3) as follows

u1 1113954ee1 minus 1113954de3 minus y2x2 + y1x1 minus h1e1

u2 1113954k y22 minus y

211113872 1113873 +1113954l x

22 minus x

211113872 1113873 minus h2e2

u3 1113954ce3 + 1113954δe1 + 1113954ρe2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(39)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimates of theparameters d e k l m c δ and ρ respectively and hi (i

1 2 3) are positive constants

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

2

1

0

ndash1

2

1

0

ndash1

2

1

0

ndash1

x(t)

y(t)

z(t)

Figure 5 Time histories of the controlled fractional order new financial system (22)

7

8

6

5

4

3

2

1

0

ndash10 1 2 3 4 5 6 7 8 9 10

Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

δhat = 1 = δlhat = 01 = l ehat = 03 = e

ρhat = 005 = ρ

Para

met

er es

timat

es

Figure 6 Parameter estimates for adaptive control in fractional order case

Mathematical Problems in Engineering 9

By replacing the control law (39) in (38) we have

_e1 minus (e minus 1113954e)e1 +(d minus 1113954d)e3 minus h1e1

_e2 minus (k minus 1113954k) y22 minus y

211113872 1113873 minus (l minus 1113954l) x

22 minus x

211113872 1113873 minus h2e2

_e3 minus (c minus 1113954c)e3 minus (δ minus 1113954δ)e1 minus (ρ minus 1113954ρ)e2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(40)

Define the estimation error of unknown parameters asfollows

ed d minus 1113954d

ee e minus 1113954e

ek k minus 1113954k

el l minus 1113954l

em m minus 1113954m

ec c minus 1113954c

eδ δ minus 1113954δ

eρ ρ minus 1113954ρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(41)

By substituting equation (41) in system (40) system (40)becomes

_e1 minus eee1 + ede3 minus h1e1

_e2 minus ek y22 minus y

211113872 1113873 minus el x

22 minus x

211113872 1113873 minus h2e2

_e3 minus ece3 minus eδe1 minus eρe2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(42)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider the quadratic Lyapunov functiondefined as follows

V 12

e21 + e

22 + e

23 + e

2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(43)

which is a positive definite function on R11

e derivative with respect to time of equation (43) givesus

_V e1 _e1 + e2 _e2 + e3 _e3 + ed _ed + ee _ee

+ ek _ek + el _el + em _em + ec _ec + eδ _eδ + eρ _eρ(44)

Using system (42) and equation (15) equation (44)becomes

_V minus h1e21 minus h2e

22 minus h3e

23 + ee minus e

21 minus _1113954e1113872 1113873

+ed e1e3 minus_1113954d1113874 1113875 + ek minus e2 y

22 minus y

211113872 1113873 minus

_1113954k1113876 1113877

+el minus e2 x22 minus x

211113872 1113873 minus

_1113954l1113876 1113877 + ec minus e23 minus _1113954c1113872 1113873

+eδ minus e3e1 minus_1113954δ1113874 1113875 + eρ minus e3e2 minus _1113954ρ1113872 1113873 minus _1113954mem

(45)

From equation (45) we deduce that the estimated pa-rameters update law is

_1113954d e1e3 + h4ed_1113954e minus e

21 + h5ee

_1113954k minus e2 y22 minus y

211113872 1113873 + h6ek

_1113954l minus e2 x22 minus x

211113872 1113873 + h7el

_1113954m h8em_1113954c minus e

23 + h9ec

_1113954δ minus e3e1 + h10eδ_1113954ρ minus e3e2 + h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(46)

in which h4 h5 h6 h7 h8 h9 h10 and h11 are positiveconstants

By replacing equation (46) in equation(45) we get

_V minus h1e21 minus h2e

22 minus h3e

23 minus h4e

2d minus h5e

2e

minus h6e2k minus h7e

2l minus h8e

2m minus h9e

2c minus h10e

2δ minus h11e

2ρ lt 0

(47)

which is a negative definite function on R11 for positiveconstants hi (i 1 11) So we have found a functionwhich verifies the Lyapunov stability theorem (Vgt 0 _Vlt 0)

us the error functions converge globally and as-ymptotically towards zero for all initial conditions esynchronization of the states of the identical systems (35)and (36) is therefore complete So we have the followingresult

Theorem 3 5e identical financial systems (35) and (36)with unknown parameters are globally and asymptoticallysynchronized for all initial conditions by the adaptive controllaw (39) where the parameters update law is given by (46) andthe hi (i 1 11) are positive constants

412 Simulation Results In this part we use the fourth-order Rungendash-Kutta algorithm to solve the two identicalfinancial systems (35) and (36) with the adaptive control law(39) and the parameters update law (46) For the simulationthe time-step h 0001 is chosen e initial conditions forthe master system are (x1(0) y1(0) z1(0)) (12 15 16)

and for the slave system (x2(0) y2(0) z2(0))

(02 05 06)e parameters of the new financial system are chosen as

follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(48)

For the hi (i 1 11) of the adaptive and updatelaws we choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) minus 1 1113954k(0) 2 1113954c(0) 05

1113954m(0) 41113954l(0) 2 1113954ρ(0) 3

1113954d(0) 7 1113954δ(0) minus 05

(49)

10 Mathematical Problems in Engineering

8

4

0

ndash4

x 1(t)

x2(t)

x1(t)

x2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(a)

0 1 2 3 4 5 6 7 8 9 10Time t

3

2

1

0

y 1(t)

y2(t)

y1(t)

y2(t)

(b)

0 1 2 3 4 5 6 7 8 9 10Time t

24

0ndash2ndash4

z 1(t)

z2(t)

z1(t)

z2(t)

(c)

0 1 2 3 4 5 6 7 8 9 10Time t

1

0

ndash1

ndash2e 1(t)

e2(t)

e 3(t)

e1(t)

e2(t)e3(t)

(d)

Figure 7 (andashc) Time evolutions of the master and slave systems state variables (x1 x2) (y1 y2) (z1 z2) respectively in integer order caseand (d) time evolution of the error functions e1(black line) e2(red line) and e3(blue line)

7

6

5

4

3

2

1

0

ndash1

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

ρhat = 005 = ρ

δhat = 1 = δlhat = 01 = l ehat = 03 = e

khat = 002 = k

Figure 8 Parameter estimates for adaptive synchronization in integer order case

Mathematical Problems in Engineering 11

By applying the adaptive control law (39) and the pa-rameter update law (46) to the new controlled financialsystem (36) the results of the numerical simulations areshown in Figures 7(a)ndash7(d) and 8 From Figures 7(a)ndash7(c) itcan be seen that the state variables of the slave and mastersystems are synchronized Figure 7(d) shows the errorsystem which eventually converges to zero Finally Figure 8shows the estimated parameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρwhich as it can be seen converge towards the real values ofthe parameters of the system ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(50)

42FractionalOrderCase In this part the identical adaptivesynchronization of the new fractional order financial systemis achieved

421 Analytical Results Let us consider the master systemas being the system described with index 1 and the slavesystem as being the system described with index 2 We havetherefore for the master system the system

Dq1x1 dz1 + y1 minus e( 1113857x1

Dq2y1 minus ky

21 minus lx

21 + m

Dq3z1 minus cz1 minus δx1 minus ρy1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(51)

and for slave system we have

Dq1x2 dz2 + y2 minus e( 1113857x2 + u1

Dq2y2 minus ky

22 minus lx

22 + m + u2

Dq3z2 minus cz2 minus δx2 minus ρy2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(52)

where u1 u2 u3 are controllers to be designed so that system(52) synchronizes with system (51) and d e k l m

c δ and ρ are the unknown constant parameters of thesystem

Let us define the error functions between the statevariables of systems (2) and (51) as follows

e1 x2 minus x1

e2 y2 minus y1

e3 z2 minus z1

(53)

From equation (53) we get the following error system

Dq1e1 minus ee1 + de3 + y2x2 minus y1x1 + u1

Dq2e2 minus k y

22 minus y

211113872 1113873 minus l x

22 minus x

211113872 1113873 + u2

Dq3e3 minus ce3 minus δe1 minus ρe2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(54)

Let us define the adaptive control functionsui (i 1 2 3)

u1 1113954ee1 minus 1113954de3 minus y2x2 + y1x1 minus h1Dq1minus 1

e1 + v1

u1 1113954k y22 minus y

211113872 1113873 +1113954l x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

u3 1113954ce3 + 1113954δe1 + 1113954ρe2 minus h3Dq3minus 1

e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(55)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimates of theparameters d e k l m c δ and ρ respectively hi (i

1 2 3) are positive constants and vi (i 1 2 3) are non-linear functions that will be designed By replacing thecontrol law (55) in (54) we get

Dq1e1 minus (e minus 1113954e)e1 +(d minus 1113954d)e3 minus h1D

q1minus 1e1 + v1

Dq2e2 minus (k minus 1113954k) y

22 minus y

211113872 1113873 minus (l minus 1113954l) x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

Dq3e3 minus (c minus 1113954c)e3 minus (δ minus 1113954δ)e1 minus (ρ minus 1113954ρ)e2 minus h3D

q3minus 1e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(56)

With the estimation error of unknown parameters de-fined by equation (41) we obtain

Dq1e1 minus eee1 + ede3 minus h1D

q1minus 1e1 + v1

Dq2e2 minus ek y

22 minus y

211113872 1113873 minus el x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

Dq3e3 minus ece3 minus eδe1 minus eρe2 minus h3D

q3minus 1e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(57)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider the quadratic Lyapunov functiondefined as follows

V 12

e21 + e

22 + e

23 + e

2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(58)

which is a positive definite function on R11 e derivativewith respect to time of equation (58) gives us

_V e1 _e1 + e2 _e2 + e3 _e3 + ed _ed + ee _ee

+ ek _ek + el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

e1D1minus q1D

q1e1 + e2D1minus q2D

q2e2

+ e3D1minus q3D

q3e3 + ed _ed + ee _ee + ek _ek

+ el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

(59)

Taking into account system (57) and equation (15)equation (59) becomes

_V e1D1minus q1 minus eee1 + ede3 minus h1D

q1minus 1e1 + v11113960 1113961

+ e2D1minus q2 minus ek y

22 minus y

211113872 1113873 minus el x

22 minus x

211113872 11138731113960

minus h2Dq2minus 1

e2 + v21113961 + e3D1minus q3 minus ece3 minus eδe11113960

minus eρe2 minus h3Dq3minus 1

e3 + v31113961 minus_ded minus _eee minus

_kek

minus_lel minus _mem minus _1113954cec minus

_δeδ minus _ρeρ

(60)

From equation (60) we deduce that the estimated pa-rameters update law is

12 Mathematical Problems in Engineering

_1113954d h4ed

_1113954e h5ee

_1113954k h6ek

_1113954l h7el

_1113954m h8em

_1113954c h9ec

_1113954δ h10eδ

_1113954ρ h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

where h4 h5 h6 h7 h8 h9 h10 and h11 are positive constantsFrom equation (60) we also deduce that the nonlinear

functions vi (i 1 2 3) are given by

v1 eee1 minus ede3

v2 ek y22 minus y

211113872 1113873 + el x

22 minus x

211113872 1113873

v3 ece3 + eδe1 + eρe2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(62)

By replacing equations (61) and (62) in (60) we get

_V minus h1e21 minus h2e

22 minus h3e

23 minus h4e

2d minus h5e

2e

minus h6e2k minus h7e

2l minus h8e

2m minus h9e

2c minus h10e

2δ

minus h11e2ρ lt 0

(63)

which is a negative definite function on R11 for positiveconstants hi (i 1 11) So we have found a functionwhich verifies the Lyapunov stability theorem (Vgt 0 _Vlt 0)

us the error functions converge globally and as-ymptotically towards zero for all initial conditions esynchronization of the states of the identical systems (51)and (52) is therefore complete So we have the followingresult

Theorem 4 5e identical financial systems (51) and (52)with unknown parameters are globally and asymptoticallysynchronized for all initial conditions by the adaptive controllaw (55) with vi (i 1 2 3) given by (62) and where theparameters update law is given by (61) and thehi (i 1 11) are positive constants

422 Simulation Results In this part we use the Adams-Bashforth-Moulton predictor-corrector method proposedby Diethelm et al [35] to solve the two identical fractionalorder systems (51) and (52) with the adaptive control law(55) the vi (i 1 2 3) given by (62) and the law forupdating the parameters is given by (61) For the simulationthe time-step h 0001 is chosen e initial conditions forthe master system are (x1(0) y1(0) z1(0)) (12 15 16)

and for the slave system (x2(0) y2(0) z2(0))

(02 05 06) are usede orders qi (i 1 2 3) are taken asfollows (q1 q2 q3) (1 088 1) ie the case of incom-mensurate order

e parameters of the new fractional order financialsystem are chosen as follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(64)

0 1 2 3 4 5 6 7 8 9 10

8

4

0

ndash4

x 1(t)

x2(t)

x1(t)

x2(t)

Time t

(a)

3

2

1

0

y 1(t)

y2(t)

y1(t)

y2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(b)

20

ndash2ndash4

z 1(t)

z2(t)

z1(t)

z2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(c)

2

1

0

ndash1e 1(t)

e2(t)

e 3(t)

e1(t)

e2(t)e3(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(d)

Figure 9 (andashc) Time evolutions of the master and slave systems state variables (x1 x2) (y1 y2) (z1 z2) respectively in fractional ordercase and (d) time evolution of the error functions e1(black line) e2(red line) and e3(blue line)

Mathematical Problems in Engineering 13

For hi (i 1 11) of the adaptive and update lawswe choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) 1 1113954k(0) minus 1 1113954c(0) 5 1113954m(0) 61113954l(0) 3 1113954ρ(0) 4 1113954d(0) 3 1113954δ(0) 7 (65)

By applying the adaptive control law (55) and the parameterupdate law (61) to the new controlled fractional-order financialsystem (52) the results of numerical simulations are shown inFigures 9(a)ndash9(d) and 10 From Figures 9(a)ndash9(c) it can be seenthat the state variables of the master and slave systems aresynchronized Figure 9(d) shows the error system whicheventually converges to zero Finally Figure 10 shows the es-timated parameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ which as it can beseen converge towards the real values of the parameters of thesystem ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(66)

5 Conclusions

In this paper the adaptive control and the adaptive syn-chronization of a new financial system with unknownconstant parameters were studied in the cases of integer andfractional order e adaptive control law and the adaptivesynchronization law were designed based on Lyapunovrsquosstability theory and on the adaptive control theory e lawshave been designed in the cases of integer and incom-mensurate fractional order system

e proposed adaptive control technique is effective forchaos control and synchronization of the new financial

system when the constant parameters of the system areunknown Numerical simulations are carried out to provethe efficiency of the control and synchronization techniquesdesigned in this work

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank IMSP-UAC and the German AcademicExchange Service (DAAD) for financial support under theprogramme ldquoIn-CountryIn-Region Scholarship Programmerdquo

References

[1] K B Oldham and J Spanier 5e Fractional Calculus Aca-demic Press New York NY USA 1974

[2] B Ross ldquoFractional calculus and its applicationsrdquo LectureNotes in Mathematics Vol 457 Springer-Verlag New YorkNY USA 1975

7

6

5

4

3

2

1

0

ndash1

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

ρhat = 005 = ρ δhat = 1 = δlhat = 1 = l ehat = 03 = e

Figure 10 Parameter estimates for adaptive synchronization in fractional order case

14 Mathematical Problems in Engineering

[3] V Daftardar-Gejji and S Bhalekar ldquoChaos in fractional or-dered Liu systemrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1117ndash1127 2010

[4] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons amp Fractals vol 36no 5 pp 1305ndash1314 2008

[5] K Rajagopal A Bayani S Jafari A Karthikeyan andI Hussain ldquoChaotic dynamics of a fractional order glucose-insulin regulatory systemrdquo Frontiers of Information Tech-nology amp Electronic Engineering vol 21 no 7 pp 1108ndash11182019

[6] T T Hartley C F Lorenzo and H Killory Qammer ldquoChaosin a fractional order Chuarsquos systemrdquo IEEE Transactions onCircuits and Systems I Fundamental 5eory and Applicationsvol 42 no 8 pp 485ndash490 1995

[7] Q Jia ldquoChaos control and synchronization of the Newton-Leipnik chaotic systemrdquo Chaos Solitons amp Fractals vol 35no 4 pp 814ndash824 2008

[8] M T Yassen ldquoAdaptive control and synchronization of amodified Chuarsquos circuit systemrdquo Applied Mathematics andComputation vol 135 no 1 pp 113ndash128 2003

[9] V Sundarapandian ldquoAdaptive control and synchronizationof a generalized Lotka-Volterra systemrdquo International Journalof Bioinformatics amp Biosciences vol 1 no 1 pp 1ndash12 2011

[10] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A StatisticalMechanics and Its Applications vol 389 no 12 pp 2434ndash2442 2010

[11] U E Kocamaz A Goksu H Taskin and Y UyarogluldquoControl of chaotic two-predator one-prey model with singlestate control signalsrdquo Journal of Intelligent Manufacturingpp 1ndash10 2020

[12] A Hajipour and H Tavakoli ldquoDynamic analysis and adaptivesliding mode controller for a chaotic fractional incommen-surate order financial systemrdquo International Journal of Bi-furcation and Chaos vol 27 no 13 p 14 2017

[13] X Yi R Guo and Y Qi ldquoStabilization of chaotic systems withboth uncertainty and disturbance by the UDE-based controlmethodrdquo IEEE Access vol 8 no 1 pp 62471ndash62477 2020

[14] L Liu B Li and R Guo ldquoConsensus control for networkedmanipulators with switched parameters and topologiesrdquo IEEEAccess vol 9 pp 9209ndash9217 2021

[15] T Hou Y Liu and F Deng ldquoFinite horizon H2∕Hinfin controlfor SDEs with infinite Markovian jumpsrdquo Nonlinear AnalysisHybrid Systems vol 34 pp 108ndash120 2019

[16] R Xu and F Zhang ldquo-Nash mean-field games for generallinear-quadratic systems with applicationsrdquo Automaticavol 114 pp 1ndash4 2020

[17] R HilferApplications of Fractional Calculus in Physics WorldScientific Hackensack NJ USA 2001

[18] R He and P G Vaidya ldquoImplementation of chaotic cryp-tography with chaotic synchronizationrdquo Physical Review Evol 57 no 2 pp 1532ndash1535 1998

[19] J H Park ldquoChaos synchronization of a chaotic system vianonlinear controlrdquo Chaos Solitons amp Fractals vol 25 no 3pp 579ndash584 2005

[20] L Huang R Feng andMWang ldquoSynchronization of chaoticsystems via nonlinear controlrdquo Physics Letters A vol 320no 4 pp 271ndash275 2004

[21] S Bhalekar and V Daftardar-Gejji ldquoSynchronization ofdifferent fractional order chaotic systems using active con-trolrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3536ndash3546 2010

[22] H N Agiza and M T Yassen ldquoSynchronization of Rosslerand Chen chaotic dynamical systems using active controlrdquoPhysics Letters A vol 278 no 4 pp 191ndash197 2001

[23] T-l Liao ldquoAdaptive synchronization of two Lorenz SystemsrdquoChaos Solitons amp Fractals vol 9 no 9 pp 1555ndash1561 1998

[24] H Yu G Cai and Y Li ldquoDynamic analysis and control of anew hyperchaotic finance systemrdquo Nonlinear Dynamicsvol 67 no 3 pp 2171ndash2182 2012

[25] J-h Ma and Y-s Chen ldquoStudy for the bifurcation topologicalstructure and the global complicated character of a kind ofnonlinear finance system Irdquo Applied Mathematics and Me-chanics vol 22 no 11 pp 1240ndash1251 2001

[26] M S Abd-Elouahab N E Hamri and J Wang ldquoChaoscontrol of a fractional order financial systemrdquo MathematicalProblems in Engineering vol 2010 Article ID 27064618 pages 2010

[27] I Hernandez C Mateos J Nuntildeez and A F Tenorio ldquoLietheory applications to problems in mathematical finance andeconomicsrdquo Applied Mathematics and Computation vol 208no 2 pp 446ndash452 2009

[28] S A David J A T Machado D D Quintino andJ M Balthazar ldquoPartial chaos suppression in a fractionalorder macroeconomic modelrdquo Mathematics and Computersin Simulation vol 122 pp 55ndash68 2016

[29] J Yang E Zhang and M Liu ldquoBifurcation analysis and chaoscontrol in a modified finance system with delayed feedbackrdquoInternational Journal of Bifurcation and Chaos vol 26 no 062016

[30] A Hajipour and H Tavakoli ldquoAnalysis and circuit simulationof a novel nonlinear fractional incommensurate order fi-nancial systemrdquo Optik vol 127 no 22 pp 10643ndash106522016

[31] X Zhao Z Li and S Li ldquoSynchronization of a chaotic financesystemrdquo Applied Mathematics and Computation vol 217no 13 pp 6031ndash6039 2011

[32] Y Liao Y Zhou F Xu and X-B Shu ldquoA study on thecomplexity of a new chaotic financial systemrdquo Complexityvol 2020 Article ID 8821156 5 pages 2020

[33] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

[34] I Petras Fractional-order Nonlinear Systems ModelingAnalysis and Simulation Springer Science amp Business MediaBerlin Germany 2011

[35] K Diethelm N J Ford andA D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquo Nonlinear Dynamics vol 29 no 1ndash4 pp 3ndash222002

Mathematical Problems in Engineering 15

_ed minus_1113954d

_ee minus _1113954e

_ek minus_1113954k _el minus

_1113954l _em minus _1113954m _ec minus _1113954c _eδ minus_1113954δ _eρ minus _1113954ρ

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

(15)

Using system (12) and equation (15) equations (14)becomes

_V minus h1x2

minus h2y2

minus h3z2

+ ed(xz minus_1113954d)

+ ee minus x2

minus _1113954e1113872 1113873 + ek minus y3

minus_1113954k1113874 1113875

+ el minus yx2

minus_1113954l1113874 1113875 + em(y minus _1113954m)

+ ec minus z2

minus _1113954c1113872 1113873 + eδ(minus xz minus_1113954δ)

+ eρ(minus zy minus _1113954ρ)

(16)

From equation (16) we deduce that the estimated pa-rameters update law is

_d xz + h4ed

_e minus x2

+ h5ee

_k minus y

3+ h6ek

_l minus yx

2+ h7el

_m y + h8em

_c minus z2

+ h9ec

_δ minus xz + h10eδ_ρ minus zy + h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

in which h4 h5 h6 h7 h8 h9 h10 and h11 are positiveconstants

By replacing equation (17) in (16) we have

_V minus h1x2

minus h2y2

minus h3z2

minus h4e2d

minus h5e2e minus h6e

2k minus h7e

2l minus h8e

2m minus h9e

2c

minus h10e2δ minus h11e

2ρ lt 0

(18)

which is the negative definite function on R11 for positiveconstants hi (i 1 11)

0ndash5ndash10 5 10x

4

3

2

1

0

ndash1

ndash2

y

(a)

0ndash5ndash10 5 10x

6

4

2

0

ndash2

ndash4

ndash6

z

(b)

500 600 700 800 900t

10

5

0

ndash5

ndash10

x

(c)

500 600 700 800 900t

4

3

2

1

0

ndash1

ndash2

y

(d)

Figure 1 Phase diagrams and time histories of system (6) (a) projected onto x-y phase plane (b) projected onto x-z phase plane (c) timehistory of (x) and (d) time history of y

4 Mathematical Problems in Engineering

So we have found a function which verifies the Lya-punov stability theorem (Vgt 0 _Vlt 0)

us we have the following result

Theorem 1 5e new financial system (8) with the unknownparameters is globally and asymptotically stabilized at theorigin for all initial conditions by the adaptive law (9) wherethe update law of the parameters is given by (17) with hi (i

1 11) being positive constants

312 Simulation Results In this part we use the fourth-order RungendashKutta algorithm to solve the new financialsystem (8) with the adaptive law (9) and the parameters

update law (17) For the simulation the time-step h 0001is chosen e initial conditions (x(0) y(0) z(0))

(12 15 16) are used e parameters of the new financialsystem are chosen as follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(19)

For hi (1 11) of the adaptive and update laws wechoose hi 3

For the initial value of the estimated parameters weassume the following values

e(0) 3 k(0) 1 1113954c(0) 4 m(0) 3 l(0) 1 ρ(0) 2 d(0) 1 δ(0) 3 (20)

By applying the adaptive control law (9) and the pa-rameter update law (17) to the new controlled financial system(8) the results of the numerical simulations are shown in

Figures 3 and 4 From Figure 3 it can be seen that the statevariables of the system converge asymptotically towards theorigin (zero) Figure 4 shows the estimated parameters for

4

3

2

1

0

ndash1

ndash20ndash5ndash10 5 10x

y

(a)

0ndash5ndash10 5 10x

z

4

2

0

ndash2

ndash4

(b)

200 300 400 500 600t

x

5

0

ndash5

ndash10

(c)

4

3

2

1

0

ndash2

ndash1

200 300 400 500 600t

y

(d)

Figure 2 Phase diagrams and time histories of system (7) with q1 1 q2 088 and q3 1 (a) projected onto x-y phase plane(b) projected onto x-z phase plane (c) time history of (x) and (d) time history of y

Mathematical Problems in Engineering 5

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

2

1

0

ndash1

2

1

0

ndash1

2

1

0

ndash1

x(t)

y(t)

z(t)

Figure 3 Time histories of the controlled integer order new financial system (8)

4

35

3

25

2

15

1

05

0

ndash05

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

δhat = 1 = δ

lhat = 01 = l

ehat = 03 = e

ρhat = 005 = ρ

Figure 4 Parameter estimates for adaptive control in integer order case

6 Mathematical Problems in Engineering

adaptive control in integer order case and as it can be seenthese parameters converge towards the real values of theparameters of the system ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(21)

32 Fractional Order Case In this part the adaptive controlof the new fractional order financial system is performed toglobally stabilize the new financial system with fractionalorder

321 Controller Design Consider the following new frac-tional order controlled financial system

Dq1x dz +(y minus e)x + u1

Dq2y minus ky

2minus lx

2+ m + u2

Dq3z minus cz minus δx minus ρy + u3

⎧⎪⎪⎨

⎪⎪⎩(22)

in which the ui (i 1 2 3) are adaptive controllers whichwill be subsequently designed taking into account the satevariables of the system and the estimation of the unknownconstant parameters d e k l m c δ and ρ of the system

To allow the states of the system to converge asymp-totically to the origin we take the following adaptive controlfunctions

u1 minus yx minus 1113954dz + 1113954ex minus h1Dq1minus 1

x + v1

u2 1113954ky2

+1113954lx2

minus 1113954m minus h2Dq2minus 1

y + v2

u3 1113954cz + 1113954δx + 1113954ρy minus h3Dq3minus 1

z + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(23)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimation of theunknown constant parametersd e k l m c δ and ρ re-spectively e hi (i 1 2 3) are positive constants andvi (i 1 2 3) are nonlinear functions that will be designed

By replacing the control law (23) in system (22) we have

Dq1x (d minus 1113954d)z minus (e minus 1113954e)x minus h1D

q1minus 1x + v1

Dq2y minus (k minus 1113954k)y

2minus (l minus 1113954l)x

2+(m minus 1113954m) minus h2D

q2minus 1y + v2

Dq3z minus (c minus 1113954c)z minus (δ minus 1113954δ)x minus (ρ minus 1113954ρ)y

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(24)

With the estimation error of unknown parameters de-fined by equation (11) we obtain

Dq1x edz minus eex minus h1D

q1minus 1x + v1

Dq2y minus eky

2minus elx

2+ em minus h2D

q2minus 1y + v2

Dq3z minus ecz minus eδx minus eρy minus h3D

q3minus 1z + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(25)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider Lyapunov quadratic function definedas follows

V 12

x2

+ y2

+ z2

+ e2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(26)

which is a positive definite function on R11e derivative with respect to time of equation (26) gives

us

_V x _x + y _y + z _z + ed _ed + ee _ee + ek _ek

+ el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

xD1minus q1D

q1x + yD1minus q2D

q2y + zD1minus q3D

q3z

+ ed _ed + ee _ee + ek _ek + el _el + em _em

+ ec _ec + eδ _eδ + eρ _eρ

(27)

Taking into account system (25) and equation (15)equation (27) becomes

_V xD1minus q1 edz minus eex minus h1D

q1minus 1x + v11113960 1113961

+ yD1minus q2 minus eky

2minus elx

2+ em minus h2D

q2minus 1y + v21113960 1113961

+ zD1minus q3 minus ecz minus eδx minus eρy minus h3D

q3minus 1z + v31113960 1113961

minus_1113954ded minus _1113954eee minus

_1113954kek minus_1113954lel minus _1113954mem minus _1113954cec

minus_1113954δeδ minus _1113954ρeρ

(28)

From equation (28) we deduce that the estimated pa-rameters update law is

_1113954d h4ed

_1113954e h5ee

_1113954k h6ek

_1113954l h7el

_1113954m h8em

_1113954c h9ec

_1113954δ h10eδ

_1113954ρ h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(29)

in which h4 h5 h6 h7 h8 h9 h10 and h11 are positiveconstants

From equation (28) we also deduce that the nonlinearfunctions vi (i 1 2 3) are given by

v1 minus edz + eex

v2 eky2

+ elx2

minus em

v3 ecz + eδx + eρy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)

By replacing equations (29) and (30) in equation (28) weget

Mathematical Problems in Engineering 7

_V minus h1x2

minus h2y2

minus h3z2

minus h4e2d

minus h5e2e minus h6e

2k minus h7e

2l minus h8e

2m minus h9e

2c

minus h10e2δ minus h11e

2ρ lt 0

(31)

which is a negative definite function on R11 for positiveconstants hi (i 1 11)

So we have found a function which verifies the Lya-punov stability theorem Vgt 0 _Vlt 0

us we have the following result

Theorem 2 5e new fractional order financial system (22)with the unknown parameters is globally and asymptoticallystabilized at the origin for all initial conditions by the adaptivelaw (23) with the vi (i 1 2 3) given by (30) and where theparameters update law is given by (29) with hi (i 1 11)

being positive constants

322 Simulation Results In this part we use the Adams-Bashforth-Moulton predictor-corrector method proposedby Diethelm et al [35] to solve the new fractional orderfinancial system (1922) with the adaptive control law (23)the vi (i 1 2 3) given by (30) and the parameter updatelaw given by (29) For the simulation the time-step h

0001 is chosen e initial conditions (x(0) y(0) z(0))

(12 15 16) are usede orders qi (i 1 2 3) are taken asfollows (q1 q2 q3) (1 088 1) ie the case of incom-mensurate order

e parameters of the new financial system are chosen asfollows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(32)

For hi (i 1 11) of the adaptive and update lawswe choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) 4 1113954k(0) 2 1113954c(0) 3 1113954m(0) 51113954l(0) 2 1113954ρ(0) 3 1113954d(0) 7 1113954δ(0) 8 (33)

By applying the adaptive law (23) and the parametersupdate law (29) to the new fractional order controlledfinancial system (22) the results of numerical simulationsare shown in Figures 5 and 6 From Figure 5 it can be seenthat the state variables of the system converge asymp-totically towards the origin Figure 6 shows the estimatedparameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ which as it can be seenconverge towards the real values of the parameters of thesystem ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(34)

4 Adaptive Synchronization of the IdenticalNew Financial System

41 Integer Order Case In this part the identical adaptivesynchronization of the new integer order financial system isachieved

411 Analytical Results Let us consider the master systemas being the system described with index 1 and the slavesystem as being the system described with index 2 We havetherefore for the master system the system

_x1 dz1 + y1 minus e( 1113857x1

_y1 minus ky21 minus lx

21 + m

_z1 minus cz1 minus δx1 minus ρy1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(35)

and for slave system we have

_x2 dz2 + y2 minus e( 1113857x2 + u1

_y2 minus ky22 minus lx

22 + m + u2

_z2 minus cz2 minus δx2 minus ρy2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(36)

in which u1 u2 u3 are controllers to be designed so thatsystem (36) synchronizes with system (35) andd e k l m c δ and ρ are the unknown constant parametersof the system

Let us define the error functions between the statevariables of systems (36) and (35) as follows

e1 x2 minus x1

e2 y2 minus y1

e3 z2 minus z1

(37)

From equation (37) we obtain the following errorsystem

_e1 minus ee1 + de3 + y2x2 minus y1x1 + u1

_e2 minus k y22 minus y

211113872 1113873 minus l x

22 minus x

211113872 1113873 + u2

_e3 minus ce3 minus δe1 minus ρe2 + u3

⎧⎪⎪⎨

⎪⎪⎩(38)

Let us define the adaptive control functionsui (i 1 2 3) as follows

u1 1113954ee1 minus 1113954de3 minus y2x2 + y1x1 minus h1e1

u2 1113954k y22 minus y

211113872 1113873 +1113954l x

22 minus x

211113872 1113873 minus h2e2

u3 1113954ce3 + 1113954δe1 + 1113954ρe2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(39)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimates of theparameters d e k l m c δ and ρ respectively and hi (i

1 2 3) are positive constants

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

2

1

0

ndash1

2

1

0

ndash1

2

1

0

ndash1

x(t)

y(t)

z(t)

Figure 5 Time histories of the controlled fractional order new financial system (22)

7

8

6

5

4

3

2

1

0

ndash10 1 2 3 4 5 6 7 8 9 10

Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

δhat = 1 = δlhat = 01 = l ehat = 03 = e

ρhat = 005 = ρ

Para

met

er es

timat

es

Figure 6 Parameter estimates for adaptive control in fractional order case

Mathematical Problems in Engineering 9

By replacing the control law (39) in (38) we have

_e1 minus (e minus 1113954e)e1 +(d minus 1113954d)e3 minus h1e1

_e2 minus (k minus 1113954k) y22 minus y

211113872 1113873 minus (l minus 1113954l) x

22 minus x

211113872 1113873 minus h2e2

_e3 minus (c minus 1113954c)e3 minus (δ minus 1113954δ)e1 minus (ρ minus 1113954ρ)e2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(40)

Define the estimation error of unknown parameters asfollows

ed d minus 1113954d

ee e minus 1113954e

ek k minus 1113954k

el l minus 1113954l

em m minus 1113954m

ec c minus 1113954c

eδ δ minus 1113954δ

eρ ρ minus 1113954ρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(41)

By substituting equation (41) in system (40) system (40)becomes

_e1 minus eee1 + ede3 minus h1e1

_e2 minus ek y22 minus y

211113872 1113873 minus el x

22 minus x

211113872 1113873 minus h2e2

_e3 minus ece3 minus eδe1 minus eρe2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(42)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider the quadratic Lyapunov functiondefined as follows

V 12

e21 + e

22 + e

23 + e

2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(43)

which is a positive definite function on R11

e derivative with respect to time of equation (43) givesus

_V e1 _e1 + e2 _e2 + e3 _e3 + ed _ed + ee _ee

+ ek _ek + el _el + em _em + ec _ec + eδ _eδ + eρ _eρ(44)

Using system (42) and equation (15) equation (44)becomes

_V minus h1e21 minus h2e

22 minus h3e

23 + ee minus e

21 minus _1113954e1113872 1113873

+ed e1e3 minus_1113954d1113874 1113875 + ek minus e2 y

22 minus y

211113872 1113873 minus

_1113954k1113876 1113877

+el minus e2 x22 minus x

211113872 1113873 minus

_1113954l1113876 1113877 + ec minus e23 minus _1113954c1113872 1113873

+eδ minus e3e1 minus_1113954δ1113874 1113875 + eρ minus e3e2 minus _1113954ρ1113872 1113873 minus _1113954mem

(45)

From equation (45) we deduce that the estimated pa-rameters update law is

_1113954d e1e3 + h4ed_1113954e minus e

21 + h5ee

_1113954k minus e2 y22 minus y

211113872 1113873 + h6ek

_1113954l minus e2 x22 minus x

211113872 1113873 + h7el

_1113954m h8em_1113954c minus e

23 + h9ec

_1113954δ minus e3e1 + h10eδ_1113954ρ minus e3e2 + h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(46)

in which h4 h5 h6 h7 h8 h9 h10 and h11 are positiveconstants

By replacing equation (46) in equation(45) we get

_V minus h1e21 minus h2e

22 minus h3e

23 minus h4e

2d minus h5e

2e

minus h6e2k minus h7e

2l minus h8e

2m minus h9e

2c minus h10e

2δ minus h11e

2ρ lt 0

(47)

which is a negative definite function on R11 for positiveconstants hi (i 1 11) So we have found a functionwhich verifies the Lyapunov stability theorem (Vgt 0 _Vlt 0)

us the error functions converge globally and as-ymptotically towards zero for all initial conditions esynchronization of the states of the identical systems (35)and (36) is therefore complete So we have the followingresult

Theorem 3 5e identical financial systems (35) and (36)with unknown parameters are globally and asymptoticallysynchronized for all initial conditions by the adaptive controllaw (39) where the parameters update law is given by (46) andthe hi (i 1 11) are positive constants

412 Simulation Results In this part we use the fourth-order Rungendash-Kutta algorithm to solve the two identicalfinancial systems (35) and (36) with the adaptive control law(39) and the parameters update law (46) For the simulationthe time-step h 0001 is chosen e initial conditions forthe master system are (x1(0) y1(0) z1(0)) (12 15 16)

and for the slave system (x2(0) y2(0) z2(0))

(02 05 06)e parameters of the new financial system are chosen as

follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(48)

For the hi (i 1 11) of the adaptive and updatelaws we choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) minus 1 1113954k(0) 2 1113954c(0) 05

1113954m(0) 41113954l(0) 2 1113954ρ(0) 3

1113954d(0) 7 1113954δ(0) minus 05

(49)

10 Mathematical Problems in Engineering

8

4

0

ndash4

x 1(t)

x2(t)

x1(t)

x2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(a)

0 1 2 3 4 5 6 7 8 9 10Time t

3

2

1

0

y 1(t)

y2(t)

y1(t)

y2(t)

(b)

0 1 2 3 4 5 6 7 8 9 10Time t

24

0ndash2ndash4

z 1(t)

z2(t)

z1(t)

z2(t)

(c)

0 1 2 3 4 5 6 7 8 9 10Time t

1

0

ndash1

ndash2e 1(t)

e2(t)

e 3(t)

e1(t)

e2(t)e3(t)

(d)

Figure 7 (andashc) Time evolutions of the master and slave systems state variables (x1 x2) (y1 y2) (z1 z2) respectively in integer order caseand (d) time evolution of the error functions e1(black line) e2(red line) and e3(blue line)

7

6

5

4

3

2

1

0

ndash1

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

ρhat = 005 = ρ

δhat = 1 = δlhat = 01 = l ehat = 03 = e

khat = 002 = k

Figure 8 Parameter estimates for adaptive synchronization in integer order case

Mathematical Problems in Engineering 11

By applying the adaptive control law (39) and the pa-rameter update law (46) to the new controlled financialsystem (36) the results of the numerical simulations areshown in Figures 7(a)ndash7(d) and 8 From Figures 7(a)ndash7(c) itcan be seen that the state variables of the slave and mastersystems are synchronized Figure 7(d) shows the errorsystem which eventually converges to zero Finally Figure 8shows the estimated parameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρwhich as it can be seen converge towards the real values ofthe parameters of the system ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(50)

42FractionalOrderCase In this part the identical adaptivesynchronization of the new fractional order financial systemis achieved

421 Analytical Results Let us consider the master systemas being the system described with index 1 and the slavesystem as being the system described with index 2 We havetherefore for the master system the system

Dq1x1 dz1 + y1 minus e( 1113857x1

Dq2y1 minus ky

21 minus lx

21 + m

Dq3z1 minus cz1 minus δx1 minus ρy1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(51)

and for slave system we have

Dq1x2 dz2 + y2 minus e( 1113857x2 + u1

Dq2y2 minus ky

22 minus lx

22 + m + u2

Dq3z2 minus cz2 minus δx2 minus ρy2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(52)

where u1 u2 u3 are controllers to be designed so that system(52) synchronizes with system (51) and d e k l m

c δ and ρ are the unknown constant parameters of thesystem

Let us define the error functions between the statevariables of systems (2) and (51) as follows

e1 x2 minus x1

e2 y2 minus y1

e3 z2 minus z1

(53)

From equation (53) we get the following error system

Dq1e1 minus ee1 + de3 + y2x2 minus y1x1 + u1

Dq2e2 minus k y

22 minus y

211113872 1113873 minus l x

22 minus x

211113872 1113873 + u2

Dq3e3 minus ce3 minus δe1 minus ρe2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(54)

Let us define the adaptive control functionsui (i 1 2 3)

u1 1113954ee1 minus 1113954de3 minus y2x2 + y1x1 minus h1Dq1minus 1

e1 + v1

u1 1113954k y22 minus y

211113872 1113873 +1113954l x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

u3 1113954ce3 + 1113954δe1 + 1113954ρe2 minus h3Dq3minus 1

e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(55)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimates of theparameters d e k l m c δ and ρ respectively hi (i

1 2 3) are positive constants and vi (i 1 2 3) are non-linear functions that will be designed By replacing thecontrol law (55) in (54) we get

Dq1e1 minus (e minus 1113954e)e1 +(d minus 1113954d)e3 minus h1D

q1minus 1e1 + v1

Dq2e2 minus (k minus 1113954k) y

22 minus y

211113872 1113873 minus (l minus 1113954l) x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

Dq3e3 minus (c minus 1113954c)e3 minus (δ minus 1113954δ)e1 minus (ρ minus 1113954ρ)e2 minus h3D

q3minus 1e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(56)

With the estimation error of unknown parameters de-fined by equation (41) we obtain

Dq1e1 minus eee1 + ede3 minus h1D

q1minus 1e1 + v1

Dq2e2 minus ek y

22 minus y

211113872 1113873 minus el x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

Dq3e3 minus ece3 minus eδe1 minus eρe2 minus h3D

q3minus 1e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(57)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider the quadratic Lyapunov functiondefined as follows

V 12

e21 + e

22 + e

23 + e

2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(58)

which is a positive definite function on R11 e derivativewith respect to time of equation (58) gives us

_V e1 _e1 + e2 _e2 + e3 _e3 + ed _ed + ee _ee

+ ek _ek + el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

e1D1minus q1D

q1e1 + e2D1minus q2D

q2e2

+ e3D1minus q3D

q3e3 + ed _ed + ee _ee + ek _ek

+ el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

(59)

Taking into account system (57) and equation (15)equation (59) becomes

_V e1D1minus q1 minus eee1 + ede3 minus h1D

q1minus 1e1 + v11113960 1113961

+ e2D1minus q2 minus ek y

22 minus y

211113872 1113873 minus el x

22 minus x

211113872 11138731113960

minus h2Dq2minus 1

e2 + v21113961 + e3D1minus q3 minus ece3 minus eδe11113960

minus eρe2 minus h3Dq3minus 1

e3 + v31113961 minus_ded minus _eee minus

_kek

minus_lel minus _mem minus _1113954cec minus

_δeδ minus _ρeρ

(60)

From equation (60) we deduce that the estimated pa-rameters update law is

12 Mathematical Problems in Engineering

_1113954d h4ed

_1113954e h5ee

_1113954k h6ek

_1113954l h7el

_1113954m h8em

_1113954c h9ec

_1113954δ h10eδ

_1113954ρ h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

where h4 h5 h6 h7 h8 h9 h10 and h11 are positive constantsFrom equation (60) we also deduce that the nonlinear

functions vi (i 1 2 3) are given by

v1 eee1 minus ede3

v2 ek y22 minus y

211113872 1113873 + el x

22 minus x

211113872 1113873

v3 ece3 + eδe1 + eρe2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(62)

By replacing equations (61) and (62) in (60) we get

_V minus h1e21 minus h2e

22 minus h3e

23 minus h4e

2d minus h5e

2e

minus h6e2k minus h7e

2l minus h8e

2m minus h9e

2c minus h10e

2δ

minus h11e2ρ lt 0

(63)

which is a negative definite function on R11 for positiveconstants hi (i 1 11) So we have found a functionwhich verifies the Lyapunov stability theorem (Vgt 0 _Vlt 0)

us the error functions converge globally and as-ymptotically towards zero for all initial conditions esynchronization of the states of the identical systems (51)and (52) is therefore complete So we have the followingresult

Theorem 4 5e identical financial systems (51) and (52)with unknown parameters are globally and asymptoticallysynchronized for all initial conditions by the adaptive controllaw (55) with vi (i 1 2 3) given by (62) and where theparameters update law is given by (61) and thehi (i 1 11) are positive constants

422 Simulation Results In this part we use the Adams-Bashforth-Moulton predictor-corrector method proposedby Diethelm et al [35] to solve the two identical fractionalorder systems (51) and (52) with the adaptive control law(55) the vi (i 1 2 3) given by (62) and the law forupdating the parameters is given by (61) For the simulationthe time-step h 0001 is chosen e initial conditions forthe master system are (x1(0) y1(0) z1(0)) (12 15 16)

and for the slave system (x2(0) y2(0) z2(0))

(02 05 06) are usede orders qi (i 1 2 3) are taken asfollows (q1 q2 q3) (1 088 1) ie the case of incom-mensurate order

e parameters of the new fractional order financialsystem are chosen as follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(64)

0 1 2 3 4 5 6 7 8 9 10

8

4

0

ndash4

x 1(t)

x2(t)

x1(t)

x2(t)

Time t

(a)

3

2

1

0

y 1(t)

y2(t)

y1(t)

y2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(b)

20

ndash2ndash4

z 1(t)

z2(t)

z1(t)

z2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(c)

2

1

0

ndash1e 1(t)

e2(t)

e 3(t)

e1(t)

e2(t)e3(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(d)

Figure 9 (andashc) Time evolutions of the master and slave systems state variables (x1 x2) (y1 y2) (z1 z2) respectively in fractional ordercase and (d) time evolution of the error functions e1(black line) e2(red line) and e3(blue line)

Mathematical Problems in Engineering 13

For hi (i 1 11) of the adaptive and update lawswe choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) 1 1113954k(0) minus 1 1113954c(0) 5 1113954m(0) 61113954l(0) 3 1113954ρ(0) 4 1113954d(0) 3 1113954δ(0) 7 (65)

By applying the adaptive control law (55) and the parameterupdate law (61) to the new controlled fractional-order financialsystem (52) the results of numerical simulations are shown inFigures 9(a)ndash9(d) and 10 From Figures 9(a)ndash9(c) it can be seenthat the state variables of the master and slave systems aresynchronized Figure 9(d) shows the error system whicheventually converges to zero Finally Figure 10 shows the es-timated parameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ which as it can beseen converge towards the real values of the parameters of thesystem ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(66)

5 Conclusions

In this paper the adaptive control and the adaptive syn-chronization of a new financial system with unknownconstant parameters were studied in the cases of integer andfractional order e adaptive control law and the adaptivesynchronization law were designed based on Lyapunovrsquosstability theory and on the adaptive control theory e lawshave been designed in the cases of integer and incom-mensurate fractional order system

e proposed adaptive control technique is effective forchaos control and synchronization of the new financial

system when the constant parameters of the system areunknown Numerical simulations are carried out to provethe efficiency of the control and synchronization techniquesdesigned in this work

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank IMSP-UAC and the German AcademicExchange Service (DAAD) for financial support under theprogramme ldquoIn-CountryIn-Region Scholarship Programmerdquo

References

[1] K B Oldham and J Spanier 5e Fractional Calculus Aca-demic Press New York NY USA 1974

[2] B Ross ldquoFractional calculus and its applicationsrdquo LectureNotes in Mathematics Vol 457 Springer-Verlag New YorkNY USA 1975

7

6

5

4

3

2

1

0

ndash1

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

ρhat = 005 = ρ δhat = 1 = δlhat = 1 = l ehat = 03 = e

Figure 10 Parameter estimates for adaptive synchronization in fractional order case

14 Mathematical Problems in Engineering

[3] V Daftardar-Gejji and S Bhalekar ldquoChaos in fractional or-dered Liu systemrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1117ndash1127 2010

[4] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons amp Fractals vol 36no 5 pp 1305ndash1314 2008

[5] K Rajagopal A Bayani S Jafari A Karthikeyan andI Hussain ldquoChaotic dynamics of a fractional order glucose-insulin regulatory systemrdquo Frontiers of Information Tech-nology amp Electronic Engineering vol 21 no 7 pp 1108ndash11182019

[6] T T Hartley C F Lorenzo and H Killory Qammer ldquoChaosin a fractional order Chuarsquos systemrdquo IEEE Transactions onCircuits and Systems I Fundamental 5eory and Applicationsvol 42 no 8 pp 485ndash490 1995

[7] Q Jia ldquoChaos control and synchronization of the Newton-Leipnik chaotic systemrdquo Chaos Solitons amp Fractals vol 35no 4 pp 814ndash824 2008

[8] M T Yassen ldquoAdaptive control and synchronization of amodified Chuarsquos circuit systemrdquo Applied Mathematics andComputation vol 135 no 1 pp 113ndash128 2003

[9] V Sundarapandian ldquoAdaptive control and synchronizationof a generalized Lotka-Volterra systemrdquo International Journalof Bioinformatics amp Biosciences vol 1 no 1 pp 1ndash12 2011

[10] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A StatisticalMechanics and Its Applications vol 389 no 12 pp 2434ndash2442 2010

[11] U E Kocamaz A Goksu H Taskin and Y UyarogluldquoControl of chaotic two-predator one-prey model with singlestate control signalsrdquo Journal of Intelligent Manufacturingpp 1ndash10 2020

[12] A Hajipour and H Tavakoli ldquoDynamic analysis and adaptivesliding mode controller for a chaotic fractional incommen-surate order financial systemrdquo International Journal of Bi-furcation and Chaos vol 27 no 13 p 14 2017

[13] X Yi R Guo and Y Qi ldquoStabilization of chaotic systems withboth uncertainty and disturbance by the UDE-based controlmethodrdquo IEEE Access vol 8 no 1 pp 62471ndash62477 2020

[14] L Liu B Li and R Guo ldquoConsensus control for networkedmanipulators with switched parameters and topologiesrdquo IEEEAccess vol 9 pp 9209ndash9217 2021

[15] T Hou Y Liu and F Deng ldquoFinite horizon H2∕Hinfin controlfor SDEs with infinite Markovian jumpsrdquo Nonlinear AnalysisHybrid Systems vol 34 pp 108ndash120 2019

[16] R Xu and F Zhang ldquo-Nash mean-field games for generallinear-quadratic systems with applicationsrdquo Automaticavol 114 pp 1ndash4 2020

[17] R HilferApplications of Fractional Calculus in Physics WorldScientific Hackensack NJ USA 2001

[18] R He and P G Vaidya ldquoImplementation of chaotic cryp-tography with chaotic synchronizationrdquo Physical Review Evol 57 no 2 pp 1532ndash1535 1998

[19] J H Park ldquoChaos synchronization of a chaotic system vianonlinear controlrdquo Chaos Solitons amp Fractals vol 25 no 3pp 579ndash584 2005

[20] L Huang R Feng andMWang ldquoSynchronization of chaoticsystems via nonlinear controlrdquo Physics Letters A vol 320no 4 pp 271ndash275 2004

[21] S Bhalekar and V Daftardar-Gejji ldquoSynchronization ofdifferent fractional order chaotic systems using active con-trolrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3536ndash3546 2010

[22] H N Agiza and M T Yassen ldquoSynchronization of Rosslerand Chen chaotic dynamical systems using active controlrdquoPhysics Letters A vol 278 no 4 pp 191ndash197 2001

[23] T-l Liao ldquoAdaptive synchronization of two Lorenz SystemsrdquoChaos Solitons amp Fractals vol 9 no 9 pp 1555ndash1561 1998

[24] H Yu G Cai and Y Li ldquoDynamic analysis and control of anew hyperchaotic finance systemrdquo Nonlinear Dynamicsvol 67 no 3 pp 2171ndash2182 2012

[25] J-h Ma and Y-s Chen ldquoStudy for the bifurcation topologicalstructure and the global complicated character of a kind ofnonlinear finance system Irdquo Applied Mathematics and Me-chanics vol 22 no 11 pp 1240ndash1251 2001

[26] M S Abd-Elouahab N E Hamri and J Wang ldquoChaoscontrol of a fractional order financial systemrdquo MathematicalProblems in Engineering vol 2010 Article ID 27064618 pages 2010

[27] I Hernandez C Mateos J Nuntildeez and A F Tenorio ldquoLietheory applications to problems in mathematical finance andeconomicsrdquo Applied Mathematics and Computation vol 208no 2 pp 446ndash452 2009

[28] S A David J A T Machado D D Quintino andJ M Balthazar ldquoPartial chaos suppression in a fractionalorder macroeconomic modelrdquo Mathematics and Computersin Simulation vol 122 pp 55ndash68 2016

[29] J Yang E Zhang and M Liu ldquoBifurcation analysis and chaoscontrol in a modified finance system with delayed feedbackrdquoInternational Journal of Bifurcation and Chaos vol 26 no 062016

[30] A Hajipour and H Tavakoli ldquoAnalysis and circuit simulationof a novel nonlinear fractional incommensurate order fi-nancial systemrdquo Optik vol 127 no 22 pp 10643ndash106522016

[31] X Zhao Z Li and S Li ldquoSynchronization of a chaotic financesystemrdquo Applied Mathematics and Computation vol 217no 13 pp 6031ndash6039 2011

[32] Y Liao Y Zhou F Xu and X-B Shu ldquoA study on thecomplexity of a new chaotic financial systemrdquo Complexityvol 2020 Article ID 8821156 5 pages 2020

[33] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

[34] I Petras Fractional-order Nonlinear Systems ModelingAnalysis and Simulation Springer Science amp Business MediaBerlin Germany 2011

[35] K Diethelm N J Ford andA D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquo Nonlinear Dynamics vol 29 no 1ndash4 pp 3ndash222002

Mathematical Problems in Engineering 15

So we have found a function which verifies the Lya-punov stability theorem (Vgt 0 _Vlt 0)

us we have the following result

Theorem 1 5e new financial system (8) with the unknownparameters is globally and asymptotically stabilized at theorigin for all initial conditions by the adaptive law (9) wherethe update law of the parameters is given by (17) with hi (i

1 11) being positive constants

312 Simulation Results In this part we use the fourth-order RungendashKutta algorithm to solve the new financialsystem (8) with the adaptive law (9) and the parameters

update law (17) For the simulation the time-step h 0001is chosen e initial conditions (x(0) y(0) z(0))

(12 15 16) are used e parameters of the new financialsystem are chosen as follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(19)

For hi (1 11) of the adaptive and update laws wechoose hi 3

For the initial value of the estimated parameters weassume the following values

e(0) 3 k(0) 1 1113954c(0) 4 m(0) 3 l(0) 1 ρ(0) 2 d(0) 1 δ(0) 3 (20)

By applying the adaptive control law (9) and the pa-rameter update law (17) to the new controlled financial system(8) the results of the numerical simulations are shown in

Figures 3 and 4 From Figure 3 it can be seen that the statevariables of the system converge asymptotically towards theorigin (zero) Figure 4 shows the estimated parameters for

4

3

2

1

0

ndash1

ndash20ndash5ndash10 5 10x

y

(a)

0ndash5ndash10 5 10x

z

4

2

0

ndash2

ndash4

(b)

200 300 400 500 600t

x

5

0

ndash5

ndash10

(c)

4

3

2

1

0

ndash2

ndash1

200 300 400 500 600t

y

(d)

Figure 2 Phase diagrams and time histories of system (7) with q1 1 q2 088 and q3 1 (a) projected onto x-y phase plane(b) projected onto x-z phase plane (c) time history of (x) and (d) time history of y

Mathematical Problems in Engineering 5

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

2

1

0

ndash1

2

1

0

ndash1

2

1

0

ndash1

x(t)

y(t)

z(t)

Figure 3 Time histories of the controlled integer order new financial system (8)

4

35

3

25

2

15

1

05

0

ndash05

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

δhat = 1 = δ

lhat = 01 = l

ehat = 03 = e

ρhat = 005 = ρ

Figure 4 Parameter estimates for adaptive control in integer order case

6 Mathematical Problems in Engineering

adaptive control in integer order case and as it can be seenthese parameters converge towards the real values of theparameters of the system ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(21)

32 Fractional Order Case In this part the adaptive controlof the new fractional order financial system is performed toglobally stabilize the new financial system with fractionalorder

321 Controller Design Consider the following new frac-tional order controlled financial system

Dq1x dz +(y minus e)x + u1

Dq2y minus ky

2minus lx

2+ m + u2

Dq3z minus cz minus δx minus ρy + u3

⎧⎪⎪⎨

⎪⎪⎩(22)

in which the ui (i 1 2 3) are adaptive controllers whichwill be subsequently designed taking into account the satevariables of the system and the estimation of the unknownconstant parameters d e k l m c δ and ρ of the system

To allow the states of the system to converge asymp-totically to the origin we take the following adaptive controlfunctions

u1 minus yx minus 1113954dz + 1113954ex minus h1Dq1minus 1

x + v1

u2 1113954ky2

+1113954lx2

minus 1113954m minus h2Dq2minus 1

y + v2

u3 1113954cz + 1113954δx + 1113954ρy minus h3Dq3minus 1

z + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(23)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimation of theunknown constant parametersd e k l m c δ and ρ re-spectively e hi (i 1 2 3) are positive constants andvi (i 1 2 3) are nonlinear functions that will be designed

By replacing the control law (23) in system (22) we have

Dq1x (d minus 1113954d)z minus (e minus 1113954e)x minus h1D

q1minus 1x + v1

Dq2y minus (k minus 1113954k)y

2minus (l minus 1113954l)x

2+(m minus 1113954m) minus h2D

q2minus 1y + v2

Dq3z minus (c minus 1113954c)z minus (δ minus 1113954δ)x minus (ρ minus 1113954ρ)y

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(24)

With the estimation error of unknown parameters de-fined by equation (11) we obtain

Dq1x edz minus eex minus h1D

q1minus 1x + v1

Dq2y minus eky

2minus elx

2+ em minus h2D

q2minus 1y + v2

Dq3z minus ecz minus eδx minus eρy minus h3D

q3minus 1z + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(25)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider Lyapunov quadratic function definedas follows

V 12

x2

+ y2

+ z2

+ e2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(26)

which is a positive definite function on R11e derivative with respect to time of equation (26) gives

us

_V x _x + y _y + z _z + ed _ed + ee _ee + ek _ek

+ el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

xD1minus q1D

q1x + yD1minus q2D

q2y + zD1minus q3D

q3z

+ ed _ed + ee _ee + ek _ek + el _el + em _em

+ ec _ec + eδ _eδ + eρ _eρ

(27)

Taking into account system (25) and equation (15)equation (27) becomes

_V xD1minus q1 edz minus eex minus h1D

q1minus 1x + v11113960 1113961

+ yD1minus q2 minus eky

2minus elx

2+ em minus h2D

q2minus 1y + v21113960 1113961

+ zD1minus q3 minus ecz minus eδx minus eρy minus h3D

q3minus 1z + v31113960 1113961

minus_1113954ded minus _1113954eee minus

_1113954kek minus_1113954lel minus _1113954mem minus _1113954cec

minus_1113954δeδ minus _1113954ρeρ

(28)

From equation (28) we deduce that the estimated pa-rameters update law is

_1113954d h4ed

_1113954e h5ee

_1113954k h6ek

_1113954l h7el

_1113954m h8em

_1113954c h9ec

_1113954δ h10eδ

_1113954ρ h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(29)

in which h4 h5 h6 h7 h8 h9 h10 and h11 are positiveconstants

From equation (28) we also deduce that the nonlinearfunctions vi (i 1 2 3) are given by

v1 minus edz + eex

v2 eky2

+ elx2

minus em

v3 ecz + eδx + eρy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)

By replacing equations (29) and (30) in equation (28) weget

Mathematical Problems in Engineering 7

_V minus h1x2

minus h2y2

minus h3z2

minus h4e2d

minus h5e2e minus h6e

2k minus h7e

2l minus h8e

2m minus h9e

2c

minus h10e2δ minus h11e

2ρ lt 0

(31)

which is a negative definite function on R11 for positiveconstants hi (i 1 11)

So we have found a function which verifies the Lya-punov stability theorem Vgt 0 _Vlt 0

us we have the following result

Theorem 2 5e new fractional order financial system (22)with the unknown parameters is globally and asymptoticallystabilized at the origin for all initial conditions by the adaptivelaw (23) with the vi (i 1 2 3) given by (30) and where theparameters update law is given by (29) with hi (i 1 11)

being positive constants

322 Simulation Results In this part we use the Adams-Bashforth-Moulton predictor-corrector method proposedby Diethelm et al [35] to solve the new fractional orderfinancial system (1922) with the adaptive control law (23)the vi (i 1 2 3) given by (30) and the parameter updatelaw given by (29) For the simulation the time-step h

0001 is chosen e initial conditions (x(0) y(0) z(0))

(12 15 16) are usede orders qi (i 1 2 3) are taken asfollows (q1 q2 q3) (1 088 1) ie the case of incom-mensurate order

e parameters of the new financial system are chosen asfollows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(32)

For hi (i 1 11) of the adaptive and update lawswe choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) 4 1113954k(0) 2 1113954c(0) 3 1113954m(0) 51113954l(0) 2 1113954ρ(0) 3 1113954d(0) 7 1113954δ(0) 8 (33)

By applying the adaptive law (23) and the parametersupdate law (29) to the new fractional order controlledfinancial system (22) the results of numerical simulationsare shown in Figures 5 and 6 From Figure 5 it can be seenthat the state variables of the system converge asymp-totically towards the origin Figure 6 shows the estimatedparameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ which as it can be seenconverge towards the real values of the parameters of thesystem ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(34)

4 Adaptive Synchronization of the IdenticalNew Financial System

41 Integer Order Case In this part the identical adaptivesynchronization of the new integer order financial system isachieved

411 Analytical Results Let us consider the master systemas being the system described with index 1 and the slavesystem as being the system described with index 2 We havetherefore for the master system the system

_x1 dz1 + y1 minus e( 1113857x1

_y1 minus ky21 minus lx

21 + m

_z1 minus cz1 minus δx1 minus ρy1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(35)

and for slave system we have

_x2 dz2 + y2 minus e( 1113857x2 + u1

_y2 minus ky22 minus lx

22 + m + u2

_z2 minus cz2 minus δx2 minus ρy2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(36)

in which u1 u2 u3 are controllers to be designed so thatsystem (36) synchronizes with system (35) andd e k l m c δ and ρ are the unknown constant parametersof the system

Let us define the error functions between the statevariables of systems (36) and (35) as follows

e1 x2 minus x1

e2 y2 minus y1

e3 z2 minus z1

(37)

From equation (37) we obtain the following errorsystem

_e1 minus ee1 + de3 + y2x2 minus y1x1 + u1

_e2 minus k y22 minus y

211113872 1113873 minus l x

22 minus x

211113872 1113873 + u2

_e3 minus ce3 minus δe1 minus ρe2 + u3

⎧⎪⎪⎨

⎪⎪⎩(38)

Let us define the adaptive control functionsui (i 1 2 3) as follows

u1 1113954ee1 minus 1113954de3 minus y2x2 + y1x1 minus h1e1

u2 1113954k y22 minus y

211113872 1113873 +1113954l x

22 minus x

211113872 1113873 minus h2e2

u3 1113954ce3 + 1113954δe1 + 1113954ρe2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(39)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimates of theparameters d e k l m c δ and ρ respectively and hi (i

1 2 3) are positive constants

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

2

1

0

ndash1

2

1

0

ndash1

2

1

0

ndash1

x(t)

y(t)

z(t)

Figure 5 Time histories of the controlled fractional order new financial system (22)

7

8

6

5

4

3

2

1

0

ndash10 1 2 3 4 5 6 7 8 9 10

Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

δhat = 1 = δlhat = 01 = l ehat = 03 = e

ρhat = 005 = ρ

Para

met

er es

timat

es

Figure 6 Parameter estimates for adaptive control in fractional order case

Mathematical Problems in Engineering 9

By replacing the control law (39) in (38) we have

_e1 minus (e minus 1113954e)e1 +(d minus 1113954d)e3 minus h1e1

_e2 minus (k minus 1113954k) y22 minus y

211113872 1113873 minus (l minus 1113954l) x

22 minus x

211113872 1113873 minus h2e2

_e3 minus (c minus 1113954c)e3 minus (δ minus 1113954δ)e1 minus (ρ minus 1113954ρ)e2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(40)

Define the estimation error of unknown parameters asfollows

ed d minus 1113954d

ee e minus 1113954e

ek k minus 1113954k

el l minus 1113954l

em m minus 1113954m

ec c minus 1113954c

eδ δ minus 1113954δ

eρ ρ minus 1113954ρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(41)

By substituting equation (41) in system (40) system (40)becomes

_e1 minus eee1 + ede3 minus h1e1

_e2 minus ek y22 minus y

211113872 1113873 minus el x

22 minus x

211113872 1113873 minus h2e2

_e3 minus ece3 minus eδe1 minus eρe2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(42)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider the quadratic Lyapunov functiondefined as follows

V 12

e21 + e

22 + e

23 + e

2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(43)

which is a positive definite function on R11

e derivative with respect to time of equation (43) givesus

_V e1 _e1 + e2 _e2 + e3 _e3 + ed _ed + ee _ee

+ ek _ek + el _el + em _em + ec _ec + eδ _eδ + eρ _eρ(44)

Using system (42) and equation (15) equation (44)becomes

_V minus h1e21 minus h2e

22 minus h3e

23 + ee minus e

21 minus _1113954e1113872 1113873

+ed e1e3 minus_1113954d1113874 1113875 + ek minus e2 y

22 minus y

211113872 1113873 minus

_1113954k1113876 1113877

+el minus e2 x22 minus x

211113872 1113873 minus

_1113954l1113876 1113877 + ec minus e23 minus _1113954c1113872 1113873

+eδ minus e3e1 minus_1113954δ1113874 1113875 + eρ minus e3e2 minus _1113954ρ1113872 1113873 minus _1113954mem

(45)

From equation (45) we deduce that the estimated pa-rameters update law is

_1113954d e1e3 + h4ed_1113954e minus e

21 + h5ee

_1113954k minus e2 y22 minus y

211113872 1113873 + h6ek

_1113954l minus e2 x22 minus x

211113872 1113873 + h7el

_1113954m h8em_1113954c minus e

23 + h9ec

_1113954δ minus e3e1 + h10eδ_1113954ρ minus e3e2 + h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(46)

in which h4 h5 h6 h7 h8 h9 h10 and h11 are positiveconstants

By replacing equation (46) in equation(45) we get

_V minus h1e21 minus h2e

22 minus h3e

23 minus h4e

2d minus h5e

2e

minus h6e2k minus h7e

2l minus h8e

2m minus h9e

2c minus h10e

2δ minus h11e

2ρ lt 0

(47)

which is a negative definite function on R11 for positiveconstants hi (i 1 11) So we have found a functionwhich verifies the Lyapunov stability theorem (Vgt 0 _Vlt 0)

us the error functions converge globally and as-ymptotically towards zero for all initial conditions esynchronization of the states of the identical systems (35)and (36) is therefore complete So we have the followingresult

Theorem 3 5e identical financial systems (35) and (36)with unknown parameters are globally and asymptoticallysynchronized for all initial conditions by the adaptive controllaw (39) where the parameters update law is given by (46) andthe hi (i 1 11) are positive constants

412 Simulation Results In this part we use the fourth-order Rungendash-Kutta algorithm to solve the two identicalfinancial systems (35) and (36) with the adaptive control law(39) and the parameters update law (46) For the simulationthe time-step h 0001 is chosen e initial conditions forthe master system are (x1(0) y1(0) z1(0)) (12 15 16)

and for the slave system (x2(0) y2(0) z2(0))

(02 05 06)e parameters of the new financial system are chosen as

follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(48)

For the hi (i 1 11) of the adaptive and updatelaws we choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) minus 1 1113954k(0) 2 1113954c(0) 05

1113954m(0) 41113954l(0) 2 1113954ρ(0) 3

1113954d(0) 7 1113954δ(0) minus 05

(49)

10 Mathematical Problems in Engineering

8

4

0

ndash4

x 1(t)

x2(t)

x1(t)

x2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(a)

0 1 2 3 4 5 6 7 8 9 10Time t

3

2

1

0

y 1(t)

y2(t)

y1(t)

y2(t)

(b)

0 1 2 3 4 5 6 7 8 9 10Time t

24

0ndash2ndash4

z 1(t)

z2(t)

z1(t)

z2(t)

(c)

0 1 2 3 4 5 6 7 8 9 10Time t

1

0

ndash1

ndash2e 1(t)

e2(t)

e 3(t)

e1(t)

e2(t)e3(t)

(d)

Figure 7 (andashc) Time evolutions of the master and slave systems state variables (x1 x2) (y1 y2) (z1 z2) respectively in integer order caseand (d) time evolution of the error functions e1(black line) e2(red line) and e3(blue line)

7

6

5

4

3

2

1

0

ndash1

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

ρhat = 005 = ρ

δhat = 1 = δlhat = 01 = l ehat = 03 = e

khat = 002 = k

Figure 8 Parameter estimates for adaptive synchronization in integer order case

Mathematical Problems in Engineering 11

By applying the adaptive control law (39) and the pa-rameter update law (46) to the new controlled financialsystem (36) the results of the numerical simulations areshown in Figures 7(a)ndash7(d) and 8 From Figures 7(a)ndash7(c) itcan be seen that the state variables of the slave and mastersystems are synchronized Figure 7(d) shows the errorsystem which eventually converges to zero Finally Figure 8shows the estimated parameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρwhich as it can be seen converge towards the real values ofthe parameters of the system ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(50)

42FractionalOrderCase In this part the identical adaptivesynchronization of the new fractional order financial systemis achieved

421 Analytical Results Let us consider the master systemas being the system described with index 1 and the slavesystem as being the system described with index 2 We havetherefore for the master system the system

Dq1x1 dz1 + y1 minus e( 1113857x1

Dq2y1 minus ky

21 minus lx

21 + m

Dq3z1 minus cz1 minus δx1 minus ρy1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(51)

and for slave system we have

Dq1x2 dz2 + y2 minus e( 1113857x2 + u1

Dq2y2 minus ky

22 minus lx

22 + m + u2

Dq3z2 minus cz2 minus δx2 minus ρy2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(52)

where u1 u2 u3 are controllers to be designed so that system(52) synchronizes with system (51) and d e k l m

c δ and ρ are the unknown constant parameters of thesystem

Let us define the error functions between the statevariables of systems (2) and (51) as follows

e1 x2 minus x1

e2 y2 minus y1

e3 z2 minus z1

(53)

From equation (53) we get the following error system

Dq1e1 minus ee1 + de3 + y2x2 minus y1x1 + u1

Dq2e2 minus k y

22 minus y

211113872 1113873 minus l x

22 minus x

211113872 1113873 + u2

Dq3e3 minus ce3 minus δe1 minus ρe2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(54)

Let us define the adaptive control functionsui (i 1 2 3)

u1 1113954ee1 minus 1113954de3 minus y2x2 + y1x1 minus h1Dq1minus 1

e1 + v1

u1 1113954k y22 minus y

211113872 1113873 +1113954l x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

u3 1113954ce3 + 1113954δe1 + 1113954ρe2 minus h3Dq3minus 1

e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(55)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimates of theparameters d e k l m c δ and ρ respectively hi (i

1 2 3) are positive constants and vi (i 1 2 3) are non-linear functions that will be designed By replacing thecontrol law (55) in (54) we get

Dq1e1 minus (e minus 1113954e)e1 +(d minus 1113954d)e3 minus h1D

q1minus 1e1 + v1

Dq2e2 minus (k minus 1113954k) y

22 minus y

211113872 1113873 minus (l minus 1113954l) x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

Dq3e3 minus (c minus 1113954c)e3 minus (δ minus 1113954δ)e1 minus (ρ minus 1113954ρ)e2 minus h3D

q3minus 1e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(56)

With the estimation error of unknown parameters de-fined by equation (41) we obtain

Dq1e1 minus eee1 + ede3 minus h1D

q1minus 1e1 + v1

Dq2e2 minus ek y

22 minus y

211113872 1113873 minus el x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

Dq3e3 minus ece3 minus eδe1 minus eρe2 minus h3D

q3minus 1e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(57)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider the quadratic Lyapunov functiondefined as follows

V 12

e21 + e

22 + e

23 + e

2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(58)

which is a positive definite function on R11 e derivativewith respect to time of equation (58) gives us

_V e1 _e1 + e2 _e2 + e3 _e3 + ed _ed + ee _ee

+ ek _ek + el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

e1D1minus q1D

q1e1 + e2D1minus q2D

q2e2

+ e3D1minus q3D

q3e3 + ed _ed + ee _ee + ek _ek

+ el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

(59)

Taking into account system (57) and equation (15)equation (59) becomes

_V e1D1minus q1 minus eee1 + ede3 minus h1D

q1minus 1e1 + v11113960 1113961

+ e2D1minus q2 minus ek y

22 minus y

211113872 1113873 minus el x

22 minus x

211113872 11138731113960

minus h2Dq2minus 1

e2 + v21113961 + e3D1minus q3 minus ece3 minus eδe11113960

minus eρe2 minus h3Dq3minus 1

e3 + v31113961 minus_ded minus _eee minus

_kek

minus_lel minus _mem minus _1113954cec minus

_δeδ minus _ρeρ

(60)

From equation (60) we deduce that the estimated pa-rameters update law is

12 Mathematical Problems in Engineering

_1113954d h4ed

_1113954e h5ee

_1113954k h6ek

_1113954l h7el

_1113954m h8em

_1113954c h9ec

_1113954δ h10eδ

_1113954ρ h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

where h4 h5 h6 h7 h8 h9 h10 and h11 are positive constantsFrom equation (60) we also deduce that the nonlinear

functions vi (i 1 2 3) are given by

v1 eee1 minus ede3

v2 ek y22 minus y

211113872 1113873 + el x

22 minus x

211113872 1113873

v3 ece3 + eδe1 + eρe2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(62)

By replacing equations (61) and (62) in (60) we get

_V minus h1e21 minus h2e

22 minus h3e

23 minus h4e

2d minus h5e

2e

minus h6e2k minus h7e

2l minus h8e

2m minus h9e

2c minus h10e

2δ

minus h11e2ρ lt 0

(63)

which is a negative definite function on R11 for positiveconstants hi (i 1 11) So we have found a functionwhich verifies the Lyapunov stability theorem (Vgt 0 _Vlt 0)

us the error functions converge globally and as-ymptotically towards zero for all initial conditions esynchronization of the states of the identical systems (51)and (52) is therefore complete So we have the followingresult

Theorem 4 5e identical financial systems (51) and (52)with unknown parameters are globally and asymptoticallysynchronized for all initial conditions by the adaptive controllaw (55) with vi (i 1 2 3) given by (62) and where theparameters update law is given by (61) and thehi (i 1 11) are positive constants

422 Simulation Results In this part we use the Adams-Bashforth-Moulton predictor-corrector method proposedby Diethelm et al [35] to solve the two identical fractionalorder systems (51) and (52) with the adaptive control law(55) the vi (i 1 2 3) given by (62) and the law forupdating the parameters is given by (61) For the simulationthe time-step h 0001 is chosen e initial conditions forthe master system are (x1(0) y1(0) z1(0)) (12 15 16)

and for the slave system (x2(0) y2(0) z2(0))

(02 05 06) are usede orders qi (i 1 2 3) are taken asfollows (q1 q2 q3) (1 088 1) ie the case of incom-mensurate order

e parameters of the new fractional order financialsystem are chosen as follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(64)

0 1 2 3 4 5 6 7 8 9 10

8

4

0

ndash4

x 1(t)

x2(t)

x1(t)

x2(t)

Time t

(a)

3

2

1

0

y 1(t)

y2(t)

y1(t)

y2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(b)

20

ndash2ndash4

z 1(t)

z2(t)

z1(t)

z2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(c)

2

1

0

ndash1e 1(t)

e2(t)

e 3(t)

e1(t)

e2(t)e3(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(d)

Figure 9 (andashc) Time evolutions of the master and slave systems state variables (x1 x2) (y1 y2) (z1 z2) respectively in fractional ordercase and (d) time evolution of the error functions e1(black line) e2(red line) and e3(blue line)

Mathematical Problems in Engineering 13

For hi (i 1 11) of the adaptive and update lawswe choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) 1 1113954k(0) minus 1 1113954c(0) 5 1113954m(0) 61113954l(0) 3 1113954ρ(0) 4 1113954d(0) 3 1113954δ(0) 7 (65)

By applying the adaptive control law (55) and the parameterupdate law (61) to the new controlled fractional-order financialsystem (52) the results of numerical simulations are shown inFigures 9(a)ndash9(d) and 10 From Figures 9(a)ndash9(c) it can be seenthat the state variables of the master and slave systems aresynchronized Figure 9(d) shows the error system whicheventually converges to zero Finally Figure 10 shows the es-timated parameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ which as it can beseen converge towards the real values of the parameters of thesystem ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(66)

5 Conclusions

In this paper the adaptive control and the adaptive syn-chronization of a new financial system with unknownconstant parameters were studied in the cases of integer andfractional order e adaptive control law and the adaptivesynchronization law were designed based on Lyapunovrsquosstability theory and on the adaptive control theory e lawshave been designed in the cases of integer and incom-mensurate fractional order system

e proposed adaptive control technique is effective forchaos control and synchronization of the new financial

system when the constant parameters of the system areunknown Numerical simulations are carried out to provethe efficiency of the control and synchronization techniquesdesigned in this work

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank IMSP-UAC and the German AcademicExchange Service (DAAD) for financial support under theprogramme ldquoIn-CountryIn-Region Scholarship Programmerdquo

References

[1] K B Oldham and J Spanier 5e Fractional Calculus Aca-demic Press New York NY USA 1974

[2] B Ross ldquoFractional calculus and its applicationsrdquo LectureNotes in Mathematics Vol 457 Springer-Verlag New YorkNY USA 1975

7

6

5

4

3

2

1

0

ndash1

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

ρhat = 005 = ρ δhat = 1 = δlhat = 1 = l ehat = 03 = e

Figure 10 Parameter estimates for adaptive synchronization in fractional order case

14 Mathematical Problems in Engineering

[3] V Daftardar-Gejji and S Bhalekar ldquoChaos in fractional or-dered Liu systemrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1117ndash1127 2010

[4] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons amp Fractals vol 36no 5 pp 1305ndash1314 2008

[5] K Rajagopal A Bayani S Jafari A Karthikeyan andI Hussain ldquoChaotic dynamics of a fractional order glucose-insulin regulatory systemrdquo Frontiers of Information Tech-nology amp Electronic Engineering vol 21 no 7 pp 1108ndash11182019

[6] T T Hartley C F Lorenzo and H Killory Qammer ldquoChaosin a fractional order Chuarsquos systemrdquo IEEE Transactions onCircuits and Systems I Fundamental 5eory and Applicationsvol 42 no 8 pp 485ndash490 1995

[7] Q Jia ldquoChaos control and synchronization of the Newton-Leipnik chaotic systemrdquo Chaos Solitons amp Fractals vol 35no 4 pp 814ndash824 2008

[8] M T Yassen ldquoAdaptive control and synchronization of amodified Chuarsquos circuit systemrdquo Applied Mathematics andComputation vol 135 no 1 pp 113ndash128 2003

[9] V Sundarapandian ldquoAdaptive control and synchronizationof a generalized Lotka-Volterra systemrdquo International Journalof Bioinformatics amp Biosciences vol 1 no 1 pp 1ndash12 2011

[10] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A StatisticalMechanics and Its Applications vol 389 no 12 pp 2434ndash2442 2010

[11] U E Kocamaz A Goksu H Taskin and Y UyarogluldquoControl of chaotic two-predator one-prey model with singlestate control signalsrdquo Journal of Intelligent Manufacturingpp 1ndash10 2020

[12] A Hajipour and H Tavakoli ldquoDynamic analysis and adaptivesliding mode controller for a chaotic fractional incommen-surate order financial systemrdquo International Journal of Bi-furcation and Chaos vol 27 no 13 p 14 2017

[13] X Yi R Guo and Y Qi ldquoStabilization of chaotic systems withboth uncertainty and disturbance by the UDE-based controlmethodrdquo IEEE Access vol 8 no 1 pp 62471ndash62477 2020

[14] L Liu B Li and R Guo ldquoConsensus control for networkedmanipulators with switched parameters and topologiesrdquo IEEEAccess vol 9 pp 9209ndash9217 2021

[15] T Hou Y Liu and F Deng ldquoFinite horizon H2∕Hinfin controlfor SDEs with infinite Markovian jumpsrdquo Nonlinear AnalysisHybrid Systems vol 34 pp 108ndash120 2019

[16] R Xu and F Zhang ldquo-Nash mean-field games for generallinear-quadratic systems with applicationsrdquo Automaticavol 114 pp 1ndash4 2020

[17] R HilferApplications of Fractional Calculus in Physics WorldScientific Hackensack NJ USA 2001

[18] R He and P G Vaidya ldquoImplementation of chaotic cryp-tography with chaotic synchronizationrdquo Physical Review Evol 57 no 2 pp 1532ndash1535 1998

[19] J H Park ldquoChaos synchronization of a chaotic system vianonlinear controlrdquo Chaos Solitons amp Fractals vol 25 no 3pp 579ndash584 2005

[20] L Huang R Feng andMWang ldquoSynchronization of chaoticsystems via nonlinear controlrdquo Physics Letters A vol 320no 4 pp 271ndash275 2004

[21] S Bhalekar and V Daftardar-Gejji ldquoSynchronization ofdifferent fractional order chaotic systems using active con-trolrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3536ndash3546 2010

[22] H N Agiza and M T Yassen ldquoSynchronization of Rosslerand Chen chaotic dynamical systems using active controlrdquoPhysics Letters A vol 278 no 4 pp 191ndash197 2001

[23] T-l Liao ldquoAdaptive synchronization of two Lorenz SystemsrdquoChaos Solitons amp Fractals vol 9 no 9 pp 1555ndash1561 1998

[24] H Yu G Cai and Y Li ldquoDynamic analysis and control of anew hyperchaotic finance systemrdquo Nonlinear Dynamicsvol 67 no 3 pp 2171ndash2182 2012

[25] J-h Ma and Y-s Chen ldquoStudy for the bifurcation topologicalstructure and the global complicated character of a kind ofnonlinear finance system Irdquo Applied Mathematics and Me-chanics vol 22 no 11 pp 1240ndash1251 2001

[26] M S Abd-Elouahab N E Hamri and J Wang ldquoChaoscontrol of a fractional order financial systemrdquo MathematicalProblems in Engineering vol 2010 Article ID 27064618 pages 2010

[27] I Hernandez C Mateos J Nuntildeez and A F Tenorio ldquoLietheory applications to problems in mathematical finance andeconomicsrdquo Applied Mathematics and Computation vol 208no 2 pp 446ndash452 2009

[28] S A David J A T Machado D D Quintino andJ M Balthazar ldquoPartial chaos suppression in a fractionalorder macroeconomic modelrdquo Mathematics and Computersin Simulation vol 122 pp 55ndash68 2016

[29] J Yang E Zhang and M Liu ldquoBifurcation analysis and chaoscontrol in a modified finance system with delayed feedbackrdquoInternational Journal of Bifurcation and Chaos vol 26 no 062016

[30] A Hajipour and H Tavakoli ldquoAnalysis and circuit simulationof a novel nonlinear fractional incommensurate order fi-nancial systemrdquo Optik vol 127 no 22 pp 10643ndash106522016

[31] X Zhao Z Li and S Li ldquoSynchronization of a chaotic financesystemrdquo Applied Mathematics and Computation vol 217no 13 pp 6031ndash6039 2011

[32] Y Liao Y Zhou F Xu and X-B Shu ldquoA study on thecomplexity of a new chaotic financial systemrdquo Complexityvol 2020 Article ID 8821156 5 pages 2020

[33] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

[34] I Petras Fractional-order Nonlinear Systems ModelingAnalysis and Simulation Springer Science amp Business MediaBerlin Germany 2011

[35] K Diethelm N J Ford andA D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquo Nonlinear Dynamics vol 29 no 1ndash4 pp 3ndash222002

Mathematical Problems in Engineering 15

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

2

1

0

ndash1

2

1

0

ndash1

2

1

0

ndash1

x(t)

y(t)

z(t)

Figure 3 Time histories of the controlled integer order new financial system (8)

4

35

3

25

2

15

1

05

0

ndash05

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

δhat = 1 = δ

lhat = 01 = l

ehat = 03 = e

ρhat = 005 = ρ

Figure 4 Parameter estimates for adaptive control in integer order case

6 Mathematical Problems in Engineering

adaptive control in integer order case and as it can be seenthese parameters converge towards the real values of theparameters of the system ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(21)

32 Fractional Order Case In this part the adaptive controlof the new fractional order financial system is performed toglobally stabilize the new financial system with fractionalorder

321 Controller Design Consider the following new frac-tional order controlled financial system

Dq1x dz +(y minus e)x + u1

Dq2y minus ky

2minus lx

2+ m + u2

Dq3z minus cz minus δx minus ρy + u3

⎧⎪⎪⎨

⎪⎪⎩(22)

in which the ui (i 1 2 3) are adaptive controllers whichwill be subsequently designed taking into account the satevariables of the system and the estimation of the unknownconstant parameters d e k l m c δ and ρ of the system

To allow the states of the system to converge asymp-totically to the origin we take the following adaptive controlfunctions

u1 minus yx minus 1113954dz + 1113954ex minus h1Dq1minus 1

x + v1

u2 1113954ky2

+1113954lx2

minus 1113954m minus h2Dq2minus 1

y + v2

u3 1113954cz + 1113954δx + 1113954ρy minus h3Dq3minus 1

z + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(23)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimation of theunknown constant parametersd e k l m c δ and ρ re-spectively e hi (i 1 2 3) are positive constants andvi (i 1 2 3) are nonlinear functions that will be designed

By replacing the control law (23) in system (22) we have

Dq1x (d minus 1113954d)z minus (e minus 1113954e)x minus h1D

q1minus 1x + v1

Dq2y minus (k minus 1113954k)y

2minus (l minus 1113954l)x

2+(m minus 1113954m) minus h2D

q2minus 1y + v2

Dq3z minus (c minus 1113954c)z minus (δ minus 1113954δ)x minus (ρ minus 1113954ρ)y

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(24)

With the estimation error of unknown parameters de-fined by equation (11) we obtain

Dq1x edz minus eex minus h1D

q1minus 1x + v1

Dq2y minus eky

2minus elx

2+ em minus h2D

q2minus 1y + v2

Dq3z minus ecz minus eδx minus eρy minus h3D

q3minus 1z + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(25)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider Lyapunov quadratic function definedas follows

V 12

x2

+ y2

+ z2

+ e2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(26)

which is a positive definite function on R11e derivative with respect to time of equation (26) gives

us

_V x _x + y _y + z _z + ed _ed + ee _ee + ek _ek

+ el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

xD1minus q1D

q1x + yD1minus q2D

q2y + zD1minus q3D

q3z

+ ed _ed + ee _ee + ek _ek + el _el + em _em

+ ec _ec + eδ _eδ + eρ _eρ

(27)

Taking into account system (25) and equation (15)equation (27) becomes

_V xD1minus q1 edz minus eex minus h1D

q1minus 1x + v11113960 1113961

+ yD1minus q2 minus eky

2minus elx

2+ em minus h2D

q2minus 1y + v21113960 1113961

+ zD1minus q3 minus ecz minus eδx minus eρy minus h3D

q3minus 1z + v31113960 1113961

minus_1113954ded minus _1113954eee minus

_1113954kek minus_1113954lel minus _1113954mem minus _1113954cec

minus_1113954δeδ minus _1113954ρeρ

(28)

From equation (28) we deduce that the estimated pa-rameters update law is

_1113954d h4ed

_1113954e h5ee

_1113954k h6ek

_1113954l h7el

_1113954m h8em

_1113954c h9ec

_1113954δ h10eδ

_1113954ρ h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(29)

in which h4 h5 h6 h7 h8 h9 h10 and h11 are positiveconstants

From equation (28) we also deduce that the nonlinearfunctions vi (i 1 2 3) are given by

v1 minus edz + eex

v2 eky2

+ elx2

minus em

v3 ecz + eδx + eρy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)

By replacing equations (29) and (30) in equation (28) weget

Mathematical Problems in Engineering 7

_V minus h1x2

minus h2y2

minus h3z2

minus h4e2d

minus h5e2e minus h6e

2k minus h7e

2l minus h8e

2m minus h9e

2c

minus h10e2δ minus h11e

2ρ lt 0

(31)

which is a negative definite function on R11 for positiveconstants hi (i 1 11)

So we have found a function which verifies the Lya-punov stability theorem Vgt 0 _Vlt 0

us we have the following result

Theorem 2 5e new fractional order financial system (22)with the unknown parameters is globally and asymptoticallystabilized at the origin for all initial conditions by the adaptivelaw (23) with the vi (i 1 2 3) given by (30) and where theparameters update law is given by (29) with hi (i 1 11)

being positive constants

322 Simulation Results In this part we use the Adams-Bashforth-Moulton predictor-corrector method proposedby Diethelm et al [35] to solve the new fractional orderfinancial system (1922) with the adaptive control law (23)the vi (i 1 2 3) given by (30) and the parameter updatelaw given by (29) For the simulation the time-step h

0001 is chosen e initial conditions (x(0) y(0) z(0))

(12 15 16) are usede orders qi (i 1 2 3) are taken asfollows (q1 q2 q3) (1 088 1) ie the case of incom-mensurate order

e parameters of the new financial system are chosen asfollows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(32)

For hi (i 1 11) of the adaptive and update lawswe choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) 4 1113954k(0) 2 1113954c(0) 3 1113954m(0) 51113954l(0) 2 1113954ρ(0) 3 1113954d(0) 7 1113954δ(0) 8 (33)

By applying the adaptive law (23) and the parametersupdate law (29) to the new fractional order controlledfinancial system (22) the results of numerical simulationsare shown in Figures 5 and 6 From Figure 5 it can be seenthat the state variables of the system converge asymp-totically towards the origin Figure 6 shows the estimatedparameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ which as it can be seenconverge towards the real values of the parameters of thesystem ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(34)

4 Adaptive Synchronization of the IdenticalNew Financial System

41 Integer Order Case In this part the identical adaptivesynchronization of the new integer order financial system isachieved

411 Analytical Results Let us consider the master systemas being the system described with index 1 and the slavesystem as being the system described with index 2 We havetherefore for the master system the system

_x1 dz1 + y1 minus e( 1113857x1

_y1 minus ky21 minus lx

21 + m

_z1 minus cz1 minus δx1 minus ρy1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(35)

and for slave system we have

_x2 dz2 + y2 minus e( 1113857x2 + u1

_y2 minus ky22 minus lx

22 + m + u2

_z2 minus cz2 minus δx2 minus ρy2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(36)

in which u1 u2 u3 are controllers to be designed so thatsystem (36) synchronizes with system (35) andd e k l m c δ and ρ are the unknown constant parametersof the system

Let us define the error functions between the statevariables of systems (36) and (35) as follows

e1 x2 minus x1

e2 y2 minus y1

e3 z2 minus z1

(37)

From equation (37) we obtain the following errorsystem

_e1 minus ee1 + de3 + y2x2 minus y1x1 + u1

_e2 minus k y22 minus y

211113872 1113873 minus l x

22 minus x

211113872 1113873 + u2

_e3 minus ce3 minus δe1 minus ρe2 + u3

⎧⎪⎪⎨

⎪⎪⎩(38)

Let us define the adaptive control functionsui (i 1 2 3) as follows

u1 1113954ee1 minus 1113954de3 minus y2x2 + y1x1 minus h1e1

u2 1113954k y22 minus y

211113872 1113873 +1113954l x

22 minus x

211113872 1113873 minus h2e2

u3 1113954ce3 + 1113954δe1 + 1113954ρe2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(39)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimates of theparameters d e k l m c δ and ρ respectively and hi (i

1 2 3) are positive constants

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

2

1

0

ndash1

2

1

0

ndash1

2

1

0

ndash1

x(t)

y(t)

z(t)

Figure 5 Time histories of the controlled fractional order new financial system (22)

7

8

6

5

4

3

2

1

0

ndash10 1 2 3 4 5 6 7 8 9 10

Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

δhat = 1 = δlhat = 01 = l ehat = 03 = e

ρhat = 005 = ρ

Para

met

er es

timat

es

Figure 6 Parameter estimates for adaptive control in fractional order case

Mathematical Problems in Engineering 9

By replacing the control law (39) in (38) we have

_e1 minus (e minus 1113954e)e1 +(d minus 1113954d)e3 minus h1e1

_e2 minus (k minus 1113954k) y22 minus y

211113872 1113873 minus (l minus 1113954l) x

22 minus x

211113872 1113873 minus h2e2

_e3 minus (c minus 1113954c)e3 minus (δ minus 1113954δ)e1 minus (ρ minus 1113954ρ)e2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(40)

Define the estimation error of unknown parameters asfollows

ed d minus 1113954d

ee e minus 1113954e

ek k minus 1113954k

el l minus 1113954l

em m minus 1113954m

ec c minus 1113954c

eδ δ minus 1113954δ

eρ ρ minus 1113954ρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(41)

By substituting equation (41) in system (40) system (40)becomes

_e1 minus eee1 + ede3 minus h1e1

_e2 minus ek y22 minus y

211113872 1113873 minus el x

22 minus x

211113872 1113873 minus h2e2

_e3 minus ece3 minus eδe1 minus eρe2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(42)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider the quadratic Lyapunov functiondefined as follows

V 12

e21 + e

22 + e

23 + e

2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(43)

which is a positive definite function on R11

e derivative with respect to time of equation (43) givesus

_V e1 _e1 + e2 _e2 + e3 _e3 + ed _ed + ee _ee

+ ek _ek + el _el + em _em + ec _ec + eδ _eδ + eρ _eρ(44)

Using system (42) and equation (15) equation (44)becomes

_V minus h1e21 minus h2e

22 minus h3e

23 + ee minus e

21 minus _1113954e1113872 1113873

+ed e1e3 minus_1113954d1113874 1113875 + ek minus e2 y

22 minus y

211113872 1113873 minus

_1113954k1113876 1113877

+el minus e2 x22 minus x

211113872 1113873 minus

_1113954l1113876 1113877 + ec minus e23 minus _1113954c1113872 1113873

+eδ minus e3e1 minus_1113954δ1113874 1113875 + eρ minus e3e2 minus _1113954ρ1113872 1113873 minus _1113954mem

(45)

From equation (45) we deduce that the estimated pa-rameters update law is

_1113954d e1e3 + h4ed_1113954e minus e

21 + h5ee

_1113954k minus e2 y22 minus y

211113872 1113873 + h6ek

_1113954l minus e2 x22 minus x

211113872 1113873 + h7el

_1113954m h8em_1113954c minus e

23 + h9ec

_1113954δ minus e3e1 + h10eδ_1113954ρ minus e3e2 + h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(46)

in which h4 h5 h6 h7 h8 h9 h10 and h11 are positiveconstants

By replacing equation (46) in equation(45) we get

_V minus h1e21 minus h2e

22 minus h3e

23 minus h4e

2d minus h5e

2e

minus h6e2k minus h7e

2l minus h8e

2m minus h9e

2c minus h10e

2δ minus h11e

2ρ lt 0

(47)

which is a negative definite function on R11 for positiveconstants hi (i 1 11) So we have found a functionwhich verifies the Lyapunov stability theorem (Vgt 0 _Vlt 0)

us the error functions converge globally and as-ymptotically towards zero for all initial conditions esynchronization of the states of the identical systems (35)and (36) is therefore complete So we have the followingresult

Theorem 3 5e identical financial systems (35) and (36)with unknown parameters are globally and asymptoticallysynchronized for all initial conditions by the adaptive controllaw (39) where the parameters update law is given by (46) andthe hi (i 1 11) are positive constants

412 Simulation Results In this part we use the fourth-order Rungendash-Kutta algorithm to solve the two identicalfinancial systems (35) and (36) with the adaptive control law(39) and the parameters update law (46) For the simulationthe time-step h 0001 is chosen e initial conditions forthe master system are (x1(0) y1(0) z1(0)) (12 15 16)

and for the slave system (x2(0) y2(0) z2(0))

(02 05 06)e parameters of the new financial system are chosen as

follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(48)

For the hi (i 1 11) of the adaptive and updatelaws we choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) minus 1 1113954k(0) 2 1113954c(0) 05

1113954m(0) 41113954l(0) 2 1113954ρ(0) 3

1113954d(0) 7 1113954δ(0) minus 05

(49)

10 Mathematical Problems in Engineering

8

4

0

ndash4

x 1(t)

x2(t)

x1(t)

x2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(a)

0 1 2 3 4 5 6 7 8 9 10Time t

3

2

1

0

y 1(t)

y2(t)

y1(t)

y2(t)

(b)

0 1 2 3 4 5 6 7 8 9 10Time t

24

0ndash2ndash4

z 1(t)

z2(t)

z1(t)

z2(t)

(c)

0 1 2 3 4 5 6 7 8 9 10Time t

1

0

ndash1

ndash2e 1(t)

e2(t)

e 3(t)

e1(t)

e2(t)e3(t)

(d)

Figure 7 (andashc) Time evolutions of the master and slave systems state variables (x1 x2) (y1 y2) (z1 z2) respectively in integer order caseand (d) time evolution of the error functions e1(black line) e2(red line) and e3(blue line)

7

6

5

4

3

2

1

0

ndash1

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

ρhat = 005 = ρ

δhat = 1 = δlhat = 01 = l ehat = 03 = e

khat = 002 = k

Figure 8 Parameter estimates for adaptive synchronization in integer order case

Mathematical Problems in Engineering 11

By applying the adaptive control law (39) and the pa-rameter update law (46) to the new controlled financialsystem (36) the results of the numerical simulations areshown in Figures 7(a)ndash7(d) and 8 From Figures 7(a)ndash7(c) itcan be seen that the state variables of the slave and mastersystems are synchronized Figure 7(d) shows the errorsystem which eventually converges to zero Finally Figure 8shows the estimated parameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρwhich as it can be seen converge towards the real values ofthe parameters of the system ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(50)

42FractionalOrderCase In this part the identical adaptivesynchronization of the new fractional order financial systemis achieved

421 Analytical Results Let us consider the master systemas being the system described with index 1 and the slavesystem as being the system described with index 2 We havetherefore for the master system the system

Dq1x1 dz1 + y1 minus e( 1113857x1

Dq2y1 minus ky

21 minus lx

21 + m

Dq3z1 minus cz1 minus δx1 minus ρy1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(51)

and for slave system we have

Dq1x2 dz2 + y2 minus e( 1113857x2 + u1

Dq2y2 minus ky

22 minus lx

22 + m + u2

Dq3z2 minus cz2 minus δx2 minus ρy2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(52)

where u1 u2 u3 are controllers to be designed so that system(52) synchronizes with system (51) and d e k l m

c δ and ρ are the unknown constant parameters of thesystem

Let us define the error functions between the statevariables of systems (2) and (51) as follows

e1 x2 minus x1

e2 y2 minus y1

e3 z2 minus z1

(53)

From equation (53) we get the following error system

Dq1e1 minus ee1 + de3 + y2x2 minus y1x1 + u1

Dq2e2 minus k y

22 minus y

211113872 1113873 minus l x

22 minus x

211113872 1113873 + u2

Dq3e3 minus ce3 minus δe1 minus ρe2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(54)

Let us define the adaptive control functionsui (i 1 2 3)

u1 1113954ee1 minus 1113954de3 minus y2x2 + y1x1 minus h1Dq1minus 1

e1 + v1

u1 1113954k y22 minus y

211113872 1113873 +1113954l x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

u3 1113954ce3 + 1113954δe1 + 1113954ρe2 minus h3Dq3minus 1

e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(55)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimates of theparameters d e k l m c δ and ρ respectively hi (i

1 2 3) are positive constants and vi (i 1 2 3) are non-linear functions that will be designed By replacing thecontrol law (55) in (54) we get

Dq1e1 minus (e minus 1113954e)e1 +(d minus 1113954d)e3 minus h1D

q1minus 1e1 + v1

Dq2e2 minus (k minus 1113954k) y

22 minus y

211113872 1113873 minus (l minus 1113954l) x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

Dq3e3 minus (c minus 1113954c)e3 minus (δ minus 1113954δ)e1 minus (ρ minus 1113954ρ)e2 minus h3D

q3minus 1e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(56)

With the estimation error of unknown parameters de-fined by equation (41) we obtain

Dq1e1 minus eee1 + ede3 minus h1D

q1minus 1e1 + v1

Dq2e2 minus ek y

22 minus y

211113872 1113873 minus el x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

Dq3e3 minus ece3 minus eδe1 minus eρe2 minus h3D

q3minus 1e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(57)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider the quadratic Lyapunov functiondefined as follows

V 12

e21 + e

22 + e

23 + e

2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(58)

which is a positive definite function on R11 e derivativewith respect to time of equation (58) gives us

_V e1 _e1 + e2 _e2 + e3 _e3 + ed _ed + ee _ee

+ ek _ek + el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

e1D1minus q1D

q1e1 + e2D1minus q2D

q2e2

+ e3D1minus q3D

q3e3 + ed _ed + ee _ee + ek _ek

+ el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

(59)

Taking into account system (57) and equation (15)equation (59) becomes

_V e1D1minus q1 minus eee1 + ede3 minus h1D

q1minus 1e1 + v11113960 1113961

+ e2D1minus q2 minus ek y

22 minus y

211113872 1113873 minus el x

22 minus x

211113872 11138731113960

minus h2Dq2minus 1

e2 + v21113961 + e3D1minus q3 minus ece3 minus eδe11113960

minus eρe2 minus h3Dq3minus 1

e3 + v31113961 minus_ded minus _eee minus

_kek

minus_lel minus _mem minus _1113954cec minus

_δeδ minus _ρeρ

(60)

From equation (60) we deduce that the estimated pa-rameters update law is

12 Mathematical Problems in Engineering

_1113954d h4ed

_1113954e h5ee

_1113954k h6ek

_1113954l h7el

_1113954m h8em

_1113954c h9ec

_1113954δ h10eδ

_1113954ρ h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

where h4 h5 h6 h7 h8 h9 h10 and h11 are positive constantsFrom equation (60) we also deduce that the nonlinear

functions vi (i 1 2 3) are given by

v1 eee1 minus ede3

v2 ek y22 minus y

211113872 1113873 + el x

22 minus x

211113872 1113873

v3 ece3 + eδe1 + eρe2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(62)

By replacing equations (61) and (62) in (60) we get

_V minus h1e21 minus h2e

22 minus h3e

23 minus h4e

2d minus h5e

2e

minus h6e2k minus h7e

2l minus h8e

2m minus h9e

2c minus h10e

2δ

minus h11e2ρ lt 0

(63)

which is a negative definite function on R11 for positiveconstants hi (i 1 11) So we have found a functionwhich verifies the Lyapunov stability theorem (Vgt 0 _Vlt 0)

us the error functions converge globally and as-ymptotically towards zero for all initial conditions esynchronization of the states of the identical systems (51)and (52) is therefore complete So we have the followingresult

Theorem 4 5e identical financial systems (51) and (52)with unknown parameters are globally and asymptoticallysynchronized for all initial conditions by the adaptive controllaw (55) with vi (i 1 2 3) given by (62) and where theparameters update law is given by (61) and thehi (i 1 11) are positive constants

422 Simulation Results In this part we use the Adams-Bashforth-Moulton predictor-corrector method proposedby Diethelm et al [35] to solve the two identical fractionalorder systems (51) and (52) with the adaptive control law(55) the vi (i 1 2 3) given by (62) and the law forupdating the parameters is given by (61) For the simulationthe time-step h 0001 is chosen e initial conditions forthe master system are (x1(0) y1(0) z1(0)) (12 15 16)

and for the slave system (x2(0) y2(0) z2(0))

(02 05 06) are usede orders qi (i 1 2 3) are taken asfollows (q1 q2 q3) (1 088 1) ie the case of incom-mensurate order

e parameters of the new fractional order financialsystem are chosen as follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(64)

0 1 2 3 4 5 6 7 8 9 10

8

4

0

ndash4

x 1(t)

x2(t)

x1(t)

x2(t)

Time t

(a)

3

2

1

0

y 1(t)

y2(t)

y1(t)

y2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(b)

20

ndash2ndash4

z 1(t)

z2(t)

z1(t)

z2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(c)

2

1

0

ndash1e 1(t)

e2(t)

e 3(t)

e1(t)

e2(t)e3(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(d)

Figure 9 (andashc) Time evolutions of the master and slave systems state variables (x1 x2) (y1 y2) (z1 z2) respectively in fractional ordercase and (d) time evolution of the error functions e1(black line) e2(red line) and e3(blue line)

Mathematical Problems in Engineering 13

For hi (i 1 11) of the adaptive and update lawswe choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) 1 1113954k(0) minus 1 1113954c(0) 5 1113954m(0) 61113954l(0) 3 1113954ρ(0) 4 1113954d(0) 3 1113954δ(0) 7 (65)

By applying the adaptive control law (55) and the parameterupdate law (61) to the new controlled fractional-order financialsystem (52) the results of numerical simulations are shown inFigures 9(a)ndash9(d) and 10 From Figures 9(a)ndash9(c) it can be seenthat the state variables of the master and slave systems aresynchronized Figure 9(d) shows the error system whicheventually converges to zero Finally Figure 10 shows the es-timated parameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ which as it can beseen converge towards the real values of the parameters of thesystem ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(66)

5 Conclusions

In this paper the adaptive control and the adaptive syn-chronization of a new financial system with unknownconstant parameters were studied in the cases of integer andfractional order e adaptive control law and the adaptivesynchronization law were designed based on Lyapunovrsquosstability theory and on the adaptive control theory e lawshave been designed in the cases of integer and incom-mensurate fractional order system

e proposed adaptive control technique is effective forchaos control and synchronization of the new financial

system when the constant parameters of the system areunknown Numerical simulations are carried out to provethe efficiency of the control and synchronization techniquesdesigned in this work

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank IMSP-UAC and the German AcademicExchange Service (DAAD) for financial support under theprogramme ldquoIn-CountryIn-Region Scholarship Programmerdquo

References

[1] K B Oldham and J Spanier 5e Fractional Calculus Aca-demic Press New York NY USA 1974

[2] B Ross ldquoFractional calculus and its applicationsrdquo LectureNotes in Mathematics Vol 457 Springer-Verlag New YorkNY USA 1975

7

6

5

4

3

2

1

0

ndash1

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

ρhat = 005 = ρ δhat = 1 = δlhat = 1 = l ehat = 03 = e

Figure 10 Parameter estimates for adaptive synchronization in fractional order case

14 Mathematical Problems in Engineering

[3] V Daftardar-Gejji and S Bhalekar ldquoChaos in fractional or-dered Liu systemrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1117ndash1127 2010

[4] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons amp Fractals vol 36no 5 pp 1305ndash1314 2008

[5] K Rajagopal A Bayani S Jafari A Karthikeyan andI Hussain ldquoChaotic dynamics of a fractional order glucose-insulin regulatory systemrdquo Frontiers of Information Tech-nology amp Electronic Engineering vol 21 no 7 pp 1108ndash11182019

[6] T T Hartley C F Lorenzo and H Killory Qammer ldquoChaosin a fractional order Chuarsquos systemrdquo IEEE Transactions onCircuits and Systems I Fundamental 5eory and Applicationsvol 42 no 8 pp 485ndash490 1995

[7] Q Jia ldquoChaos control and synchronization of the Newton-Leipnik chaotic systemrdquo Chaos Solitons amp Fractals vol 35no 4 pp 814ndash824 2008

[8] M T Yassen ldquoAdaptive control and synchronization of amodified Chuarsquos circuit systemrdquo Applied Mathematics andComputation vol 135 no 1 pp 113ndash128 2003

[9] V Sundarapandian ldquoAdaptive control and synchronizationof a generalized Lotka-Volterra systemrdquo International Journalof Bioinformatics amp Biosciences vol 1 no 1 pp 1ndash12 2011

[10] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A StatisticalMechanics and Its Applications vol 389 no 12 pp 2434ndash2442 2010

[11] U E Kocamaz A Goksu H Taskin and Y UyarogluldquoControl of chaotic two-predator one-prey model with singlestate control signalsrdquo Journal of Intelligent Manufacturingpp 1ndash10 2020

[12] A Hajipour and H Tavakoli ldquoDynamic analysis and adaptivesliding mode controller for a chaotic fractional incommen-surate order financial systemrdquo International Journal of Bi-furcation and Chaos vol 27 no 13 p 14 2017

[13] X Yi R Guo and Y Qi ldquoStabilization of chaotic systems withboth uncertainty and disturbance by the UDE-based controlmethodrdquo IEEE Access vol 8 no 1 pp 62471ndash62477 2020

[14] L Liu B Li and R Guo ldquoConsensus control for networkedmanipulators with switched parameters and topologiesrdquo IEEEAccess vol 9 pp 9209ndash9217 2021

[15] T Hou Y Liu and F Deng ldquoFinite horizon H2∕Hinfin controlfor SDEs with infinite Markovian jumpsrdquo Nonlinear AnalysisHybrid Systems vol 34 pp 108ndash120 2019

[16] R Xu and F Zhang ldquo-Nash mean-field games for generallinear-quadratic systems with applicationsrdquo Automaticavol 114 pp 1ndash4 2020

[17] R HilferApplications of Fractional Calculus in Physics WorldScientific Hackensack NJ USA 2001

[18] R He and P G Vaidya ldquoImplementation of chaotic cryp-tography with chaotic synchronizationrdquo Physical Review Evol 57 no 2 pp 1532ndash1535 1998

[19] J H Park ldquoChaos synchronization of a chaotic system vianonlinear controlrdquo Chaos Solitons amp Fractals vol 25 no 3pp 579ndash584 2005

[20] L Huang R Feng andMWang ldquoSynchronization of chaoticsystems via nonlinear controlrdquo Physics Letters A vol 320no 4 pp 271ndash275 2004

[21] S Bhalekar and V Daftardar-Gejji ldquoSynchronization ofdifferent fractional order chaotic systems using active con-trolrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3536ndash3546 2010

[22] H N Agiza and M T Yassen ldquoSynchronization of Rosslerand Chen chaotic dynamical systems using active controlrdquoPhysics Letters A vol 278 no 4 pp 191ndash197 2001

[23] T-l Liao ldquoAdaptive synchronization of two Lorenz SystemsrdquoChaos Solitons amp Fractals vol 9 no 9 pp 1555ndash1561 1998

[24] H Yu G Cai and Y Li ldquoDynamic analysis and control of anew hyperchaotic finance systemrdquo Nonlinear Dynamicsvol 67 no 3 pp 2171ndash2182 2012

[25] J-h Ma and Y-s Chen ldquoStudy for the bifurcation topologicalstructure and the global complicated character of a kind ofnonlinear finance system Irdquo Applied Mathematics and Me-chanics vol 22 no 11 pp 1240ndash1251 2001

[26] M S Abd-Elouahab N E Hamri and J Wang ldquoChaoscontrol of a fractional order financial systemrdquo MathematicalProblems in Engineering vol 2010 Article ID 27064618 pages 2010

[27] I Hernandez C Mateos J Nuntildeez and A F Tenorio ldquoLietheory applications to problems in mathematical finance andeconomicsrdquo Applied Mathematics and Computation vol 208no 2 pp 446ndash452 2009

[28] S A David J A T Machado D D Quintino andJ M Balthazar ldquoPartial chaos suppression in a fractionalorder macroeconomic modelrdquo Mathematics and Computersin Simulation vol 122 pp 55ndash68 2016

[29] J Yang E Zhang and M Liu ldquoBifurcation analysis and chaoscontrol in a modified finance system with delayed feedbackrdquoInternational Journal of Bifurcation and Chaos vol 26 no 062016

[30] A Hajipour and H Tavakoli ldquoAnalysis and circuit simulationof a novel nonlinear fractional incommensurate order fi-nancial systemrdquo Optik vol 127 no 22 pp 10643ndash106522016

[31] X Zhao Z Li and S Li ldquoSynchronization of a chaotic financesystemrdquo Applied Mathematics and Computation vol 217no 13 pp 6031ndash6039 2011

[32] Y Liao Y Zhou F Xu and X-B Shu ldquoA study on thecomplexity of a new chaotic financial systemrdquo Complexityvol 2020 Article ID 8821156 5 pages 2020

[33] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

[34] I Petras Fractional-order Nonlinear Systems ModelingAnalysis and Simulation Springer Science amp Business MediaBerlin Germany 2011

[35] K Diethelm N J Ford andA D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquo Nonlinear Dynamics vol 29 no 1ndash4 pp 3ndash222002

Mathematical Problems in Engineering 15

adaptive control in integer order case and as it can be seenthese parameters converge towards the real values of theparameters of the system ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(21)

32 Fractional Order Case In this part the adaptive controlof the new fractional order financial system is performed toglobally stabilize the new financial system with fractionalorder

321 Controller Design Consider the following new frac-tional order controlled financial system

Dq1x dz +(y minus e)x + u1

Dq2y minus ky

2minus lx

2+ m + u2

Dq3z minus cz minus δx minus ρy + u3

⎧⎪⎪⎨

⎪⎪⎩(22)

in which the ui (i 1 2 3) are adaptive controllers whichwill be subsequently designed taking into account the satevariables of the system and the estimation of the unknownconstant parameters d e k l m c δ and ρ of the system

To allow the states of the system to converge asymp-totically to the origin we take the following adaptive controlfunctions

u1 minus yx minus 1113954dz + 1113954ex minus h1Dq1minus 1

x + v1

u2 1113954ky2

+1113954lx2

minus 1113954m minus h2Dq2minus 1

y + v2

u3 1113954cz + 1113954δx + 1113954ρy minus h3Dq3minus 1

z + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(23)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimation of theunknown constant parametersd e k l m c δ and ρ re-spectively e hi (i 1 2 3) are positive constants andvi (i 1 2 3) are nonlinear functions that will be designed

By replacing the control law (23) in system (22) we have

Dq1x (d minus 1113954d)z minus (e minus 1113954e)x minus h1D

q1minus 1x + v1

Dq2y minus (k minus 1113954k)y

2minus (l minus 1113954l)x

2+(m minus 1113954m) minus h2D

q2minus 1y + v2

Dq3z minus (c minus 1113954c)z minus (δ minus 1113954δ)x minus (ρ minus 1113954ρ)y

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(24)

With the estimation error of unknown parameters de-fined by equation (11) we obtain

Dq1x edz minus eex minus h1D

q1minus 1x + v1

Dq2y minus eky

2minus elx

2+ em minus h2D

q2minus 1y + v2

Dq3z minus ecz minus eδx minus eρy minus h3D

q3minus 1z + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(25)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider Lyapunov quadratic function definedas follows

V 12

x2

+ y2

+ z2

+ e2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(26)

which is a positive definite function on R11e derivative with respect to time of equation (26) gives

us

_V x _x + y _y + z _z + ed _ed + ee _ee + ek _ek

+ el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

xD1minus q1D

q1x + yD1minus q2D

q2y + zD1minus q3D

q3z

+ ed _ed + ee _ee + ek _ek + el _el + em _em

+ ec _ec + eδ _eδ + eρ _eρ

(27)

Taking into account system (25) and equation (15)equation (27) becomes

_V xD1minus q1 edz minus eex minus h1D

q1minus 1x + v11113960 1113961

+ yD1minus q2 minus eky

2minus elx

2+ em minus h2D

q2minus 1y + v21113960 1113961

+ zD1minus q3 minus ecz minus eδx minus eρy minus h3D

q3minus 1z + v31113960 1113961

minus_1113954ded minus _1113954eee minus

_1113954kek minus_1113954lel minus _1113954mem minus _1113954cec

minus_1113954δeδ minus _1113954ρeρ

(28)

From equation (28) we deduce that the estimated pa-rameters update law is

_1113954d h4ed

_1113954e h5ee

_1113954k h6ek

_1113954l h7el

_1113954m h8em

_1113954c h9ec

_1113954δ h10eδ

_1113954ρ h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(29)

in which h4 h5 h6 h7 h8 h9 h10 and h11 are positiveconstants

From equation (28) we also deduce that the nonlinearfunctions vi (i 1 2 3) are given by

v1 minus edz + eex

v2 eky2

+ elx2

minus em

v3 ecz + eδx + eρy

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(30)

By replacing equations (29) and (30) in equation (28) weget

Mathematical Problems in Engineering 7

_V minus h1x2

minus h2y2

minus h3z2

minus h4e2d

minus h5e2e minus h6e

2k minus h7e

2l minus h8e

2m minus h9e

2c

minus h10e2δ minus h11e

2ρ lt 0

(31)

which is a negative definite function on R11 for positiveconstants hi (i 1 11)

So we have found a function which verifies the Lya-punov stability theorem Vgt 0 _Vlt 0

us we have the following result

Theorem 2 5e new fractional order financial system (22)with the unknown parameters is globally and asymptoticallystabilized at the origin for all initial conditions by the adaptivelaw (23) with the vi (i 1 2 3) given by (30) and where theparameters update law is given by (29) with hi (i 1 11)

being positive constants

322 Simulation Results In this part we use the Adams-Bashforth-Moulton predictor-corrector method proposedby Diethelm et al [35] to solve the new fractional orderfinancial system (1922) with the adaptive control law (23)the vi (i 1 2 3) given by (30) and the parameter updatelaw given by (29) For the simulation the time-step h

0001 is chosen e initial conditions (x(0) y(0) z(0))

(12 15 16) are usede orders qi (i 1 2 3) are taken asfollows (q1 q2 q3) (1 088 1) ie the case of incom-mensurate order

e parameters of the new financial system are chosen asfollows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(32)

For hi (i 1 11) of the adaptive and update lawswe choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) 4 1113954k(0) 2 1113954c(0) 3 1113954m(0) 51113954l(0) 2 1113954ρ(0) 3 1113954d(0) 7 1113954δ(0) 8 (33)

By applying the adaptive law (23) and the parametersupdate law (29) to the new fractional order controlledfinancial system (22) the results of numerical simulationsare shown in Figures 5 and 6 From Figure 5 it can be seenthat the state variables of the system converge asymp-totically towards the origin Figure 6 shows the estimatedparameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ which as it can be seenconverge towards the real values of the parameters of thesystem ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(34)

4 Adaptive Synchronization of the IdenticalNew Financial System

41 Integer Order Case In this part the identical adaptivesynchronization of the new integer order financial system isachieved

411 Analytical Results Let us consider the master systemas being the system described with index 1 and the slavesystem as being the system described with index 2 We havetherefore for the master system the system

_x1 dz1 + y1 minus e( 1113857x1

_y1 minus ky21 minus lx

21 + m

_z1 minus cz1 minus δx1 minus ρy1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(35)

and for slave system we have

_x2 dz2 + y2 minus e( 1113857x2 + u1

_y2 minus ky22 minus lx

22 + m + u2

_z2 minus cz2 minus δx2 minus ρy2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(36)

in which u1 u2 u3 are controllers to be designed so thatsystem (36) synchronizes with system (35) andd e k l m c δ and ρ are the unknown constant parametersof the system

Let us define the error functions between the statevariables of systems (36) and (35) as follows

e1 x2 minus x1

e2 y2 minus y1

e3 z2 minus z1

(37)

From equation (37) we obtain the following errorsystem

_e1 minus ee1 + de3 + y2x2 minus y1x1 + u1

_e2 minus k y22 minus y

211113872 1113873 minus l x

22 minus x

211113872 1113873 + u2

_e3 minus ce3 minus δe1 minus ρe2 + u3

⎧⎪⎪⎨

⎪⎪⎩(38)

Let us define the adaptive control functionsui (i 1 2 3) as follows

u1 1113954ee1 minus 1113954de3 minus y2x2 + y1x1 minus h1e1

u2 1113954k y22 minus y

211113872 1113873 +1113954l x

22 minus x

211113872 1113873 minus h2e2

u3 1113954ce3 + 1113954δe1 + 1113954ρe2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(39)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimates of theparameters d e k l m c δ and ρ respectively and hi (i

1 2 3) are positive constants

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

2

1

0

ndash1

2

1

0

ndash1

2

1

0

ndash1

x(t)

y(t)

z(t)

Figure 5 Time histories of the controlled fractional order new financial system (22)

7

8

6

5

4

3

2

1

0

ndash10 1 2 3 4 5 6 7 8 9 10

Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

δhat = 1 = δlhat = 01 = l ehat = 03 = e

ρhat = 005 = ρ

Para

met

er es

timat

es

Figure 6 Parameter estimates for adaptive control in fractional order case

Mathematical Problems in Engineering 9

By replacing the control law (39) in (38) we have

_e1 minus (e minus 1113954e)e1 +(d minus 1113954d)e3 minus h1e1

_e2 minus (k minus 1113954k) y22 minus y

211113872 1113873 minus (l minus 1113954l) x

22 minus x

211113872 1113873 minus h2e2

_e3 minus (c minus 1113954c)e3 minus (δ minus 1113954δ)e1 minus (ρ minus 1113954ρ)e2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(40)

Define the estimation error of unknown parameters asfollows

ed d minus 1113954d

ee e minus 1113954e

ek k minus 1113954k

el l minus 1113954l

em m minus 1113954m

ec c minus 1113954c

eδ δ minus 1113954δ

eρ ρ minus 1113954ρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(41)

By substituting equation (41) in system (40) system (40)becomes

_e1 minus eee1 + ede3 minus h1e1

_e2 minus ek y22 minus y

211113872 1113873 minus el x

22 minus x

211113872 1113873 minus h2e2

_e3 minus ece3 minus eδe1 minus eρe2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(42)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider the quadratic Lyapunov functiondefined as follows

V 12

e21 + e

22 + e

23 + e

2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(43)

which is a positive definite function on R11

e derivative with respect to time of equation (43) givesus

_V e1 _e1 + e2 _e2 + e3 _e3 + ed _ed + ee _ee

+ ek _ek + el _el + em _em + ec _ec + eδ _eδ + eρ _eρ(44)

Using system (42) and equation (15) equation (44)becomes

_V minus h1e21 minus h2e

22 minus h3e

23 + ee minus e

21 minus _1113954e1113872 1113873

+ed e1e3 minus_1113954d1113874 1113875 + ek minus e2 y

22 minus y

211113872 1113873 minus

_1113954k1113876 1113877

+el minus e2 x22 minus x

211113872 1113873 minus

_1113954l1113876 1113877 + ec minus e23 minus _1113954c1113872 1113873

+eδ minus e3e1 minus_1113954δ1113874 1113875 + eρ minus e3e2 minus _1113954ρ1113872 1113873 minus _1113954mem

(45)

From equation (45) we deduce that the estimated pa-rameters update law is

_1113954d e1e3 + h4ed_1113954e minus e

21 + h5ee

_1113954k minus e2 y22 minus y

211113872 1113873 + h6ek

_1113954l minus e2 x22 minus x

211113872 1113873 + h7el

_1113954m h8em_1113954c minus e

23 + h9ec

_1113954δ minus e3e1 + h10eδ_1113954ρ minus e3e2 + h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(46)

in which h4 h5 h6 h7 h8 h9 h10 and h11 are positiveconstants

By replacing equation (46) in equation(45) we get

_V minus h1e21 minus h2e

22 minus h3e

23 minus h4e

2d minus h5e

2e

minus h6e2k minus h7e

2l minus h8e

2m minus h9e

2c minus h10e

2δ minus h11e

2ρ lt 0

(47)

which is a negative definite function on R11 for positiveconstants hi (i 1 11) So we have found a functionwhich verifies the Lyapunov stability theorem (Vgt 0 _Vlt 0)

us the error functions converge globally and as-ymptotically towards zero for all initial conditions esynchronization of the states of the identical systems (35)and (36) is therefore complete So we have the followingresult

Theorem 3 5e identical financial systems (35) and (36)with unknown parameters are globally and asymptoticallysynchronized for all initial conditions by the adaptive controllaw (39) where the parameters update law is given by (46) andthe hi (i 1 11) are positive constants

412 Simulation Results In this part we use the fourth-order Rungendash-Kutta algorithm to solve the two identicalfinancial systems (35) and (36) with the adaptive control law(39) and the parameters update law (46) For the simulationthe time-step h 0001 is chosen e initial conditions forthe master system are (x1(0) y1(0) z1(0)) (12 15 16)

and for the slave system (x2(0) y2(0) z2(0))

(02 05 06)e parameters of the new financial system are chosen as

follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(48)

For the hi (i 1 11) of the adaptive and updatelaws we choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) minus 1 1113954k(0) 2 1113954c(0) 05

1113954m(0) 41113954l(0) 2 1113954ρ(0) 3

1113954d(0) 7 1113954δ(0) minus 05

(49)

10 Mathematical Problems in Engineering

8

4

0

ndash4

x 1(t)

x2(t)

x1(t)

x2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(a)

0 1 2 3 4 5 6 7 8 9 10Time t

3

2

1

0

y 1(t)

y2(t)

y1(t)

y2(t)

(b)

0 1 2 3 4 5 6 7 8 9 10Time t

24

0ndash2ndash4

z 1(t)

z2(t)

z1(t)

z2(t)

(c)

0 1 2 3 4 5 6 7 8 9 10Time t

1

0

ndash1

ndash2e 1(t)

e2(t)

e 3(t)

e1(t)

e2(t)e3(t)

(d)

Figure 7 (andashc) Time evolutions of the master and slave systems state variables (x1 x2) (y1 y2) (z1 z2) respectively in integer order caseand (d) time evolution of the error functions e1(black line) e2(red line) and e3(blue line)

7

6

5

4

3

2

1

0

ndash1

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

ρhat = 005 = ρ

δhat = 1 = δlhat = 01 = l ehat = 03 = e

khat = 002 = k

Figure 8 Parameter estimates for adaptive synchronization in integer order case

Mathematical Problems in Engineering 11

By applying the adaptive control law (39) and the pa-rameter update law (46) to the new controlled financialsystem (36) the results of the numerical simulations areshown in Figures 7(a)ndash7(d) and 8 From Figures 7(a)ndash7(c) itcan be seen that the state variables of the slave and mastersystems are synchronized Figure 7(d) shows the errorsystem which eventually converges to zero Finally Figure 8shows the estimated parameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρwhich as it can be seen converge towards the real values ofthe parameters of the system ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(50)

42FractionalOrderCase In this part the identical adaptivesynchronization of the new fractional order financial systemis achieved

421 Analytical Results Let us consider the master systemas being the system described with index 1 and the slavesystem as being the system described with index 2 We havetherefore for the master system the system

Dq1x1 dz1 + y1 minus e( 1113857x1

Dq2y1 minus ky

21 minus lx

21 + m

Dq3z1 minus cz1 minus δx1 minus ρy1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(51)

and for slave system we have

Dq1x2 dz2 + y2 minus e( 1113857x2 + u1

Dq2y2 minus ky

22 minus lx

22 + m + u2

Dq3z2 minus cz2 minus δx2 minus ρy2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(52)

where u1 u2 u3 are controllers to be designed so that system(52) synchronizes with system (51) and d e k l m

c δ and ρ are the unknown constant parameters of thesystem

Let us define the error functions between the statevariables of systems (2) and (51) as follows

e1 x2 minus x1

e2 y2 minus y1

e3 z2 minus z1

(53)

From equation (53) we get the following error system

Dq1e1 minus ee1 + de3 + y2x2 minus y1x1 + u1

Dq2e2 minus k y

22 minus y

211113872 1113873 minus l x

22 minus x

211113872 1113873 + u2

Dq3e3 minus ce3 minus δe1 minus ρe2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(54)

Let us define the adaptive control functionsui (i 1 2 3)

u1 1113954ee1 minus 1113954de3 minus y2x2 + y1x1 minus h1Dq1minus 1

e1 + v1

u1 1113954k y22 minus y

211113872 1113873 +1113954l x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

u3 1113954ce3 + 1113954δe1 + 1113954ρe2 minus h3Dq3minus 1

e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(55)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimates of theparameters d e k l m c δ and ρ respectively hi (i

1 2 3) are positive constants and vi (i 1 2 3) are non-linear functions that will be designed By replacing thecontrol law (55) in (54) we get

Dq1e1 minus (e minus 1113954e)e1 +(d minus 1113954d)e3 minus h1D

q1minus 1e1 + v1

Dq2e2 minus (k minus 1113954k) y

22 minus y

211113872 1113873 minus (l minus 1113954l) x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

Dq3e3 minus (c minus 1113954c)e3 minus (δ minus 1113954δ)e1 minus (ρ minus 1113954ρ)e2 minus h3D

q3minus 1e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(56)

With the estimation error of unknown parameters de-fined by equation (41) we obtain

Dq1e1 minus eee1 + ede3 minus h1D

q1minus 1e1 + v1

Dq2e2 minus ek y

22 minus y

211113872 1113873 minus el x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

Dq3e3 minus ece3 minus eδe1 minus eρe2 minus h3D

q3minus 1e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(57)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider the quadratic Lyapunov functiondefined as follows

V 12

e21 + e

22 + e

23 + e

2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(58)

which is a positive definite function on R11 e derivativewith respect to time of equation (58) gives us

_V e1 _e1 + e2 _e2 + e3 _e3 + ed _ed + ee _ee

+ ek _ek + el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

e1D1minus q1D

q1e1 + e2D1minus q2D

q2e2

+ e3D1minus q3D

q3e3 + ed _ed + ee _ee + ek _ek

+ el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

(59)

Taking into account system (57) and equation (15)equation (59) becomes

_V e1D1minus q1 minus eee1 + ede3 minus h1D

q1minus 1e1 + v11113960 1113961

+ e2D1minus q2 minus ek y

22 minus y

211113872 1113873 minus el x

22 minus x

211113872 11138731113960

minus h2Dq2minus 1

e2 + v21113961 + e3D1minus q3 minus ece3 minus eδe11113960

minus eρe2 minus h3Dq3minus 1

e3 + v31113961 minus_ded minus _eee minus

_kek

minus_lel minus _mem minus _1113954cec minus

_δeδ minus _ρeρ

(60)

From equation (60) we deduce that the estimated pa-rameters update law is

12 Mathematical Problems in Engineering

_1113954d h4ed

_1113954e h5ee

_1113954k h6ek

_1113954l h7el

_1113954m h8em

_1113954c h9ec

_1113954δ h10eδ

_1113954ρ h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

where h4 h5 h6 h7 h8 h9 h10 and h11 are positive constantsFrom equation (60) we also deduce that the nonlinear

functions vi (i 1 2 3) are given by

v1 eee1 minus ede3

v2 ek y22 minus y

211113872 1113873 + el x

22 minus x

211113872 1113873

v3 ece3 + eδe1 + eρe2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(62)

By replacing equations (61) and (62) in (60) we get

_V minus h1e21 minus h2e

22 minus h3e

23 minus h4e

2d minus h5e

2e

minus h6e2k minus h7e

2l minus h8e

2m minus h9e

2c minus h10e

2δ

minus h11e2ρ lt 0

(63)

which is a negative definite function on R11 for positiveconstants hi (i 1 11) So we have found a functionwhich verifies the Lyapunov stability theorem (Vgt 0 _Vlt 0)

us the error functions converge globally and as-ymptotically towards zero for all initial conditions esynchronization of the states of the identical systems (51)and (52) is therefore complete So we have the followingresult

Theorem 4 5e identical financial systems (51) and (52)with unknown parameters are globally and asymptoticallysynchronized for all initial conditions by the adaptive controllaw (55) with vi (i 1 2 3) given by (62) and where theparameters update law is given by (61) and thehi (i 1 11) are positive constants

422 Simulation Results In this part we use the Adams-Bashforth-Moulton predictor-corrector method proposedby Diethelm et al [35] to solve the two identical fractionalorder systems (51) and (52) with the adaptive control law(55) the vi (i 1 2 3) given by (62) and the law forupdating the parameters is given by (61) For the simulationthe time-step h 0001 is chosen e initial conditions forthe master system are (x1(0) y1(0) z1(0)) (12 15 16)

and for the slave system (x2(0) y2(0) z2(0))

(02 05 06) are usede orders qi (i 1 2 3) are taken asfollows (q1 q2 q3) (1 088 1) ie the case of incom-mensurate order

e parameters of the new fractional order financialsystem are chosen as follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(64)

0 1 2 3 4 5 6 7 8 9 10

8

4

0

ndash4

x 1(t)

x2(t)

x1(t)

x2(t)

Time t

(a)

3

2

1

0

y 1(t)

y2(t)

y1(t)

y2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(b)

20

ndash2ndash4

z 1(t)

z2(t)

z1(t)

z2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(c)

2

1

0

ndash1e 1(t)

e2(t)

e 3(t)

e1(t)

e2(t)e3(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(d)

Figure 9 (andashc) Time evolutions of the master and slave systems state variables (x1 x2) (y1 y2) (z1 z2) respectively in fractional ordercase and (d) time evolution of the error functions e1(black line) e2(red line) and e3(blue line)

Mathematical Problems in Engineering 13

For hi (i 1 11) of the adaptive and update lawswe choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) 1 1113954k(0) minus 1 1113954c(0) 5 1113954m(0) 61113954l(0) 3 1113954ρ(0) 4 1113954d(0) 3 1113954δ(0) 7 (65)

By applying the adaptive control law (55) and the parameterupdate law (61) to the new controlled fractional-order financialsystem (52) the results of numerical simulations are shown inFigures 9(a)ndash9(d) and 10 From Figures 9(a)ndash9(c) it can be seenthat the state variables of the master and slave systems aresynchronized Figure 9(d) shows the error system whicheventually converges to zero Finally Figure 10 shows the es-timated parameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ which as it can beseen converge towards the real values of the parameters of thesystem ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(66)

5 Conclusions

In this paper the adaptive control and the adaptive syn-chronization of a new financial system with unknownconstant parameters were studied in the cases of integer andfractional order e adaptive control law and the adaptivesynchronization law were designed based on Lyapunovrsquosstability theory and on the adaptive control theory e lawshave been designed in the cases of integer and incom-mensurate fractional order system

e proposed adaptive control technique is effective forchaos control and synchronization of the new financial

system when the constant parameters of the system areunknown Numerical simulations are carried out to provethe efficiency of the control and synchronization techniquesdesigned in this work

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank IMSP-UAC and the German AcademicExchange Service (DAAD) for financial support under theprogramme ldquoIn-CountryIn-Region Scholarship Programmerdquo

References

[1] K B Oldham and J Spanier 5e Fractional Calculus Aca-demic Press New York NY USA 1974

[2] B Ross ldquoFractional calculus and its applicationsrdquo LectureNotes in Mathematics Vol 457 Springer-Verlag New YorkNY USA 1975

7

6

5

4

3

2

1

0

ndash1

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

ρhat = 005 = ρ δhat = 1 = δlhat = 1 = l ehat = 03 = e

Figure 10 Parameter estimates for adaptive synchronization in fractional order case

14 Mathematical Problems in Engineering

[3] V Daftardar-Gejji and S Bhalekar ldquoChaos in fractional or-dered Liu systemrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1117ndash1127 2010

[4] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons amp Fractals vol 36no 5 pp 1305ndash1314 2008

[5] K Rajagopal A Bayani S Jafari A Karthikeyan andI Hussain ldquoChaotic dynamics of a fractional order glucose-insulin regulatory systemrdquo Frontiers of Information Tech-nology amp Electronic Engineering vol 21 no 7 pp 1108ndash11182019

[6] T T Hartley C F Lorenzo and H Killory Qammer ldquoChaosin a fractional order Chuarsquos systemrdquo IEEE Transactions onCircuits and Systems I Fundamental 5eory and Applicationsvol 42 no 8 pp 485ndash490 1995

[7] Q Jia ldquoChaos control and synchronization of the Newton-Leipnik chaotic systemrdquo Chaos Solitons amp Fractals vol 35no 4 pp 814ndash824 2008

[8] M T Yassen ldquoAdaptive control and synchronization of amodified Chuarsquos circuit systemrdquo Applied Mathematics andComputation vol 135 no 1 pp 113ndash128 2003

[9] V Sundarapandian ldquoAdaptive control and synchronizationof a generalized Lotka-Volterra systemrdquo International Journalof Bioinformatics amp Biosciences vol 1 no 1 pp 1ndash12 2011

[10] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A StatisticalMechanics and Its Applications vol 389 no 12 pp 2434ndash2442 2010

[11] U E Kocamaz A Goksu H Taskin and Y UyarogluldquoControl of chaotic two-predator one-prey model with singlestate control signalsrdquo Journal of Intelligent Manufacturingpp 1ndash10 2020

[12] A Hajipour and H Tavakoli ldquoDynamic analysis and adaptivesliding mode controller for a chaotic fractional incommen-surate order financial systemrdquo International Journal of Bi-furcation and Chaos vol 27 no 13 p 14 2017

[13] X Yi R Guo and Y Qi ldquoStabilization of chaotic systems withboth uncertainty and disturbance by the UDE-based controlmethodrdquo IEEE Access vol 8 no 1 pp 62471ndash62477 2020

[14] L Liu B Li and R Guo ldquoConsensus control for networkedmanipulators with switched parameters and topologiesrdquo IEEEAccess vol 9 pp 9209ndash9217 2021

[15] T Hou Y Liu and F Deng ldquoFinite horizon H2∕Hinfin controlfor SDEs with infinite Markovian jumpsrdquo Nonlinear AnalysisHybrid Systems vol 34 pp 108ndash120 2019

[16] R Xu and F Zhang ldquo-Nash mean-field games for generallinear-quadratic systems with applicationsrdquo Automaticavol 114 pp 1ndash4 2020

[17] R HilferApplications of Fractional Calculus in Physics WorldScientific Hackensack NJ USA 2001

[18] R He and P G Vaidya ldquoImplementation of chaotic cryp-tography with chaotic synchronizationrdquo Physical Review Evol 57 no 2 pp 1532ndash1535 1998

[19] J H Park ldquoChaos synchronization of a chaotic system vianonlinear controlrdquo Chaos Solitons amp Fractals vol 25 no 3pp 579ndash584 2005

[20] L Huang R Feng andMWang ldquoSynchronization of chaoticsystems via nonlinear controlrdquo Physics Letters A vol 320no 4 pp 271ndash275 2004

[21] S Bhalekar and V Daftardar-Gejji ldquoSynchronization ofdifferent fractional order chaotic systems using active con-trolrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3536ndash3546 2010

[22] H N Agiza and M T Yassen ldquoSynchronization of Rosslerand Chen chaotic dynamical systems using active controlrdquoPhysics Letters A vol 278 no 4 pp 191ndash197 2001

[23] T-l Liao ldquoAdaptive synchronization of two Lorenz SystemsrdquoChaos Solitons amp Fractals vol 9 no 9 pp 1555ndash1561 1998

[24] H Yu G Cai and Y Li ldquoDynamic analysis and control of anew hyperchaotic finance systemrdquo Nonlinear Dynamicsvol 67 no 3 pp 2171ndash2182 2012

[25] J-h Ma and Y-s Chen ldquoStudy for the bifurcation topologicalstructure and the global complicated character of a kind ofnonlinear finance system Irdquo Applied Mathematics and Me-chanics vol 22 no 11 pp 1240ndash1251 2001

[26] M S Abd-Elouahab N E Hamri and J Wang ldquoChaoscontrol of a fractional order financial systemrdquo MathematicalProblems in Engineering vol 2010 Article ID 27064618 pages 2010

[27] I Hernandez C Mateos J Nuntildeez and A F Tenorio ldquoLietheory applications to problems in mathematical finance andeconomicsrdquo Applied Mathematics and Computation vol 208no 2 pp 446ndash452 2009

[28] S A David J A T Machado D D Quintino andJ M Balthazar ldquoPartial chaos suppression in a fractionalorder macroeconomic modelrdquo Mathematics and Computersin Simulation vol 122 pp 55ndash68 2016

[29] J Yang E Zhang and M Liu ldquoBifurcation analysis and chaoscontrol in a modified finance system with delayed feedbackrdquoInternational Journal of Bifurcation and Chaos vol 26 no 062016

[30] A Hajipour and H Tavakoli ldquoAnalysis and circuit simulationof a novel nonlinear fractional incommensurate order fi-nancial systemrdquo Optik vol 127 no 22 pp 10643ndash106522016

[31] X Zhao Z Li and S Li ldquoSynchronization of a chaotic financesystemrdquo Applied Mathematics and Computation vol 217no 13 pp 6031ndash6039 2011

[32] Y Liao Y Zhou F Xu and X-B Shu ldquoA study on thecomplexity of a new chaotic financial systemrdquo Complexityvol 2020 Article ID 8821156 5 pages 2020

[33] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

[34] I Petras Fractional-order Nonlinear Systems ModelingAnalysis and Simulation Springer Science amp Business MediaBerlin Germany 2011

[35] K Diethelm N J Ford andA D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquo Nonlinear Dynamics vol 29 no 1ndash4 pp 3ndash222002

Mathematical Problems in Engineering 15

_V minus h1x2

minus h2y2

minus h3z2

minus h4e2d

minus h5e2e minus h6e

2k minus h7e

2l minus h8e

2m minus h9e

2c

minus h10e2δ minus h11e

2ρ lt 0

(31)

which is a negative definite function on R11 for positiveconstants hi (i 1 11)

So we have found a function which verifies the Lya-punov stability theorem Vgt 0 _Vlt 0

us we have the following result

Theorem 2 5e new fractional order financial system (22)with the unknown parameters is globally and asymptoticallystabilized at the origin for all initial conditions by the adaptivelaw (23) with the vi (i 1 2 3) given by (30) and where theparameters update law is given by (29) with hi (i 1 11)

being positive constants

322 Simulation Results In this part we use the Adams-Bashforth-Moulton predictor-corrector method proposedby Diethelm et al [35] to solve the new fractional orderfinancial system (1922) with the adaptive control law (23)the vi (i 1 2 3) given by (30) and the parameter updatelaw given by (29) For the simulation the time-step h

0001 is chosen e initial conditions (x(0) y(0) z(0))

(12 15 16) are usede orders qi (i 1 2 3) are taken asfollows (q1 q2 q3) (1 088 1) ie the case of incom-mensurate order

e parameters of the new financial system are chosen asfollows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(32)

For hi (i 1 11) of the adaptive and update lawswe choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) 4 1113954k(0) 2 1113954c(0) 3 1113954m(0) 51113954l(0) 2 1113954ρ(0) 3 1113954d(0) 7 1113954δ(0) 8 (33)

By applying the adaptive law (23) and the parametersupdate law (29) to the new fractional order controlledfinancial system (22) the results of numerical simulationsare shown in Figures 5 and 6 From Figure 5 it can be seenthat the state variables of the system converge asymp-totically towards the origin Figure 6 shows the estimatedparameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ which as it can be seenconverge towards the real values of the parameters of thesystem ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(34)

4 Adaptive Synchronization of the IdenticalNew Financial System

41 Integer Order Case In this part the identical adaptivesynchronization of the new integer order financial system isachieved

411 Analytical Results Let us consider the master systemas being the system described with index 1 and the slavesystem as being the system described with index 2 We havetherefore for the master system the system

_x1 dz1 + y1 minus e( 1113857x1

_y1 minus ky21 minus lx

21 + m

_z1 minus cz1 minus δx1 minus ρy1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(35)

and for slave system we have

_x2 dz2 + y2 minus e( 1113857x2 + u1

_y2 minus ky22 minus lx

22 + m + u2

_z2 minus cz2 minus δx2 minus ρy2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(36)

in which u1 u2 u3 are controllers to be designed so thatsystem (36) synchronizes with system (35) andd e k l m c δ and ρ are the unknown constant parametersof the system

Let us define the error functions between the statevariables of systems (36) and (35) as follows

e1 x2 minus x1

e2 y2 minus y1

e3 z2 minus z1

(37)

From equation (37) we obtain the following errorsystem

_e1 minus ee1 + de3 + y2x2 minus y1x1 + u1

_e2 minus k y22 minus y

211113872 1113873 minus l x

22 minus x

211113872 1113873 + u2

_e3 minus ce3 minus δe1 minus ρe2 + u3

⎧⎪⎪⎨

⎪⎪⎩(38)

Let us define the adaptive control functionsui (i 1 2 3) as follows

u1 1113954ee1 minus 1113954de3 minus y2x2 + y1x1 minus h1e1

u2 1113954k y22 minus y

211113872 1113873 +1113954l x

22 minus x

211113872 1113873 minus h2e2

u3 1113954ce3 + 1113954δe1 + 1113954ρe2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(39)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimates of theparameters d e k l m c δ and ρ respectively and hi (i

1 2 3) are positive constants

8 Mathematical Problems in Engineering

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

2

1

0

ndash1

2

1

0

ndash1

2

1

0

ndash1

x(t)

y(t)

z(t)

Figure 5 Time histories of the controlled fractional order new financial system (22)

7

8

6

5

4

3

2

1

0

ndash10 1 2 3 4 5 6 7 8 9 10

Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

δhat = 1 = δlhat = 01 = l ehat = 03 = e

ρhat = 005 = ρ

Para

met

er es

timat

es

Figure 6 Parameter estimates for adaptive control in fractional order case

Mathematical Problems in Engineering 9

By replacing the control law (39) in (38) we have

_e1 minus (e minus 1113954e)e1 +(d minus 1113954d)e3 minus h1e1

_e2 minus (k minus 1113954k) y22 minus y

211113872 1113873 minus (l minus 1113954l) x

22 minus x

211113872 1113873 minus h2e2

_e3 minus (c minus 1113954c)e3 minus (δ minus 1113954δ)e1 minus (ρ minus 1113954ρ)e2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(40)

Define the estimation error of unknown parameters asfollows

ed d minus 1113954d

ee e minus 1113954e

ek k minus 1113954k

el l minus 1113954l

em m minus 1113954m

ec c minus 1113954c

eδ δ minus 1113954δ

eρ ρ minus 1113954ρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(41)

By substituting equation (41) in system (40) system (40)becomes

_e1 minus eee1 + ede3 minus h1e1

_e2 minus ek y22 minus y

211113872 1113873 minus el x

22 minus x

211113872 1113873 minus h2e2

_e3 minus ece3 minus eδe1 minus eρe2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(42)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider the quadratic Lyapunov functiondefined as follows

V 12

e21 + e

22 + e

23 + e

2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(43)

which is a positive definite function on R11

e derivative with respect to time of equation (43) givesus

_V e1 _e1 + e2 _e2 + e3 _e3 + ed _ed + ee _ee

+ ek _ek + el _el + em _em + ec _ec + eδ _eδ + eρ _eρ(44)

Using system (42) and equation (15) equation (44)becomes

_V minus h1e21 minus h2e

22 minus h3e

23 + ee minus e

21 minus _1113954e1113872 1113873

+ed e1e3 minus_1113954d1113874 1113875 + ek minus e2 y

22 minus y

211113872 1113873 minus

_1113954k1113876 1113877

+el minus e2 x22 minus x

211113872 1113873 minus

_1113954l1113876 1113877 + ec minus e23 minus _1113954c1113872 1113873

+eδ minus e3e1 minus_1113954δ1113874 1113875 + eρ minus e3e2 minus _1113954ρ1113872 1113873 minus _1113954mem

(45)

From equation (45) we deduce that the estimated pa-rameters update law is

_1113954d e1e3 + h4ed_1113954e minus e

21 + h5ee

_1113954k minus e2 y22 minus y

211113872 1113873 + h6ek

_1113954l minus e2 x22 minus x

211113872 1113873 + h7el

_1113954m h8em_1113954c minus e

23 + h9ec

_1113954δ minus e3e1 + h10eδ_1113954ρ minus e3e2 + h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(46)

in which h4 h5 h6 h7 h8 h9 h10 and h11 are positiveconstants

By replacing equation (46) in equation(45) we get

_V minus h1e21 minus h2e

22 minus h3e

23 minus h4e

2d minus h5e

2e

minus h6e2k minus h7e

2l minus h8e

2m minus h9e

2c minus h10e

2δ minus h11e

2ρ lt 0

(47)

which is a negative definite function on R11 for positiveconstants hi (i 1 11) So we have found a functionwhich verifies the Lyapunov stability theorem (Vgt 0 _Vlt 0)

us the error functions converge globally and as-ymptotically towards zero for all initial conditions esynchronization of the states of the identical systems (35)and (36) is therefore complete So we have the followingresult

Theorem 3 5e identical financial systems (35) and (36)with unknown parameters are globally and asymptoticallysynchronized for all initial conditions by the adaptive controllaw (39) where the parameters update law is given by (46) andthe hi (i 1 11) are positive constants

412 Simulation Results In this part we use the fourth-order Rungendash-Kutta algorithm to solve the two identicalfinancial systems (35) and (36) with the adaptive control law(39) and the parameters update law (46) For the simulationthe time-step h 0001 is chosen e initial conditions forthe master system are (x1(0) y1(0) z1(0)) (12 15 16)

and for the slave system (x2(0) y2(0) z2(0))

(02 05 06)e parameters of the new financial system are chosen as

follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(48)

For the hi (i 1 11) of the adaptive and updatelaws we choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) minus 1 1113954k(0) 2 1113954c(0) 05

1113954m(0) 41113954l(0) 2 1113954ρ(0) 3

1113954d(0) 7 1113954δ(0) minus 05

(49)

10 Mathematical Problems in Engineering

8

4

0

ndash4

x 1(t)

x2(t)

x1(t)

x2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(a)

0 1 2 3 4 5 6 7 8 9 10Time t

3

2

1

0

y 1(t)

y2(t)

y1(t)

y2(t)

(b)

0 1 2 3 4 5 6 7 8 9 10Time t

24

0ndash2ndash4

z 1(t)

z2(t)

z1(t)

z2(t)

(c)

0 1 2 3 4 5 6 7 8 9 10Time t

1

0

ndash1

ndash2e 1(t)

e2(t)

e 3(t)

e1(t)

e2(t)e3(t)

(d)

Figure 7 (andashc) Time evolutions of the master and slave systems state variables (x1 x2) (y1 y2) (z1 z2) respectively in integer order caseand (d) time evolution of the error functions e1(black line) e2(red line) and e3(blue line)

7

6

5

4

3

2

1

0

ndash1

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

ρhat = 005 = ρ

δhat = 1 = δlhat = 01 = l ehat = 03 = e

khat = 002 = k

Figure 8 Parameter estimates for adaptive synchronization in integer order case

Mathematical Problems in Engineering 11

By applying the adaptive control law (39) and the pa-rameter update law (46) to the new controlled financialsystem (36) the results of the numerical simulations areshown in Figures 7(a)ndash7(d) and 8 From Figures 7(a)ndash7(c) itcan be seen that the state variables of the slave and mastersystems are synchronized Figure 7(d) shows the errorsystem which eventually converges to zero Finally Figure 8shows the estimated parameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρwhich as it can be seen converge towards the real values ofthe parameters of the system ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(50)

42FractionalOrderCase In this part the identical adaptivesynchronization of the new fractional order financial systemis achieved

421 Analytical Results Let us consider the master systemas being the system described with index 1 and the slavesystem as being the system described with index 2 We havetherefore for the master system the system

Dq1x1 dz1 + y1 minus e( 1113857x1

Dq2y1 minus ky

21 minus lx

21 + m

Dq3z1 minus cz1 minus δx1 minus ρy1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(51)

and for slave system we have

Dq1x2 dz2 + y2 minus e( 1113857x2 + u1

Dq2y2 minus ky

22 minus lx

22 + m + u2

Dq3z2 minus cz2 minus δx2 minus ρy2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(52)

where u1 u2 u3 are controllers to be designed so that system(52) synchronizes with system (51) and d e k l m

c δ and ρ are the unknown constant parameters of thesystem

Let us define the error functions between the statevariables of systems (2) and (51) as follows

e1 x2 minus x1

e2 y2 minus y1

e3 z2 minus z1

(53)

From equation (53) we get the following error system

Dq1e1 minus ee1 + de3 + y2x2 minus y1x1 + u1

Dq2e2 minus k y

22 minus y

211113872 1113873 minus l x

22 minus x

211113872 1113873 + u2

Dq3e3 minus ce3 minus δe1 minus ρe2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(54)

Let us define the adaptive control functionsui (i 1 2 3)

u1 1113954ee1 minus 1113954de3 minus y2x2 + y1x1 minus h1Dq1minus 1

e1 + v1

u1 1113954k y22 minus y

211113872 1113873 +1113954l x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

u3 1113954ce3 + 1113954δe1 + 1113954ρe2 minus h3Dq3minus 1

e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(55)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimates of theparameters d e k l m c δ and ρ respectively hi (i

1 2 3) are positive constants and vi (i 1 2 3) are non-linear functions that will be designed By replacing thecontrol law (55) in (54) we get

Dq1e1 minus (e minus 1113954e)e1 +(d minus 1113954d)e3 minus h1D

q1minus 1e1 + v1

Dq2e2 minus (k minus 1113954k) y

22 minus y

211113872 1113873 minus (l minus 1113954l) x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

Dq3e3 minus (c minus 1113954c)e3 minus (δ minus 1113954δ)e1 minus (ρ minus 1113954ρ)e2 minus h3D

q3minus 1e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(56)

With the estimation error of unknown parameters de-fined by equation (41) we obtain

Dq1e1 minus eee1 + ede3 minus h1D

q1minus 1e1 + v1

Dq2e2 minus ek y

22 minus y

211113872 1113873 minus el x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

Dq3e3 minus ece3 minus eδe1 minus eρe2 minus h3D

q3minus 1e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(57)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider the quadratic Lyapunov functiondefined as follows

V 12

e21 + e

22 + e

23 + e

2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(58)

which is a positive definite function on R11 e derivativewith respect to time of equation (58) gives us

_V e1 _e1 + e2 _e2 + e3 _e3 + ed _ed + ee _ee

+ ek _ek + el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

e1D1minus q1D

q1e1 + e2D1minus q2D

q2e2

+ e3D1minus q3D

q3e3 + ed _ed + ee _ee + ek _ek

+ el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

(59)

Taking into account system (57) and equation (15)equation (59) becomes

_V e1D1minus q1 minus eee1 + ede3 minus h1D

q1minus 1e1 + v11113960 1113961

+ e2D1minus q2 minus ek y

22 minus y

211113872 1113873 minus el x

22 minus x

211113872 11138731113960

minus h2Dq2minus 1

e2 + v21113961 + e3D1minus q3 minus ece3 minus eδe11113960

minus eρe2 minus h3Dq3minus 1

e3 + v31113961 minus_ded minus _eee minus

_kek

minus_lel minus _mem minus _1113954cec minus

_δeδ minus _ρeρ

(60)

From equation (60) we deduce that the estimated pa-rameters update law is

12 Mathematical Problems in Engineering

_1113954d h4ed

_1113954e h5ee

_1113954k h6ek

_1113954l h7el

_1113954m h8em

_1113954c h9ec

_1113954δ h10eδ

_1113954ρ h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

where h4 h5 h6 h7 h8 h9 h10 and h11 are positive constantsFrom equation (60) we also deduce that the nonlinear

functions vi (i 1 2 3) are given by

v1 eee1 minus ede3

v2 ek y22 minus y

211113872 1113873 + el x

22 minus x

211113872 1113873

v3 ece3 + eδe1 + eρe2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(62)

By replacing equations (61) and (62) in (60) we get

_V minus h1e21 minus h2e

22 minus h3e

23 minus h4e

2d minus h5e

2e

minus h6e2k minus h7e

2l minus h8e

2m minus h9e

2c minus h10e

2δ

minus h11e2ρ lt 0

(63)

which is a negative definite function on R11 for positiveconstants hi (i 1 11) So we have found a functionwhich verifies the Lyapunov stability theorem (Vgt 0 _Vlt 0)

us the error functions converge globally and as-ymptotically towards zero for all initial conditions esynchronization of the states of the identical systems (51)and (52) is therefore complete So we have the followingresult

Theorem 4 5e identical financial systems (51) and (52)with unknown parameters are globally and asymptoticallysynchronized for all initial conditions by the adaptive controllaw (55) with vi (i 1 2 3) given by (62) and where theparameters update law is given by (61) and thehi (i 1 11) are positive constants

422 Simulation Results In this part we use the Adams-Bashforth-Moulton predictor-corrector method proposedby Diethelm et al [35] to solve the two identical fractionalorder systems (51) and (52) with the adaptive control law(55) the vi (i 1 2 3) given by (62) and the law forupdating the parameters is given by (61) For the simulationthe time-step h 0001 is chosen e initial conditions forthe master system are (x1(0) y1(0) z1(0)) (12 15 16)

and for the slave system (x2(0) y2(0) z2(0))

(02 05 06) are usede orders qi (i 1 2 3) are taken asfollows (q1 q2 q3) (1 088 1) ie the case of incom-mensurate order

e parameters of the new fractional order financialsystem are chosen as follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(64)

0 1 2 3 4 5 6 7 8 9 10

8

4

0

ndash4

x 1(t)

x2(t)

x1(t)

x2(t)

Time t

(a)

3

2

1

0

y 1(t)

y2(t)

y1(t)

y2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(b)

20

ndash2ndash4

z 1(t)

z2(t)

z1(t)

z2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(c)

2

1

0

ndash1e 1(t)

e2(t)

e 3(t)

e1(t)

e2(t)e3(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(d)

Figure 9 (andashc) Time evolutions of the master and slave systems state variables (x1 x2) (y1 y2) (z1 z2) respectively in fractional ordercase and (d) time evolution of the error functions e1(black line) e2(red line) and e3(blue line)

Mathematical Problems in Engineering 13

For hi (i 1 11) of the adaptive and update lawswe choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) 1 1113954k(0) minus 1 1113954c(0) 5 1113954m(0) 61113954l(0) 3 1113954ρ(0) 4 1113954d(0) 3 1113954δ(0) 7 (65)

By applying the adaptive control law (55) and the parameterupdate law (61) to the new controlled fractional-order financialsystem (52) the results of numerical simulations are shown inFigures 9(a)ndash9(d) and 10 From Figures 9(a)ndash9(c) it can be seenthat the state variables of the master and slave systems aresynchronized Figure 9(d) shows the error system whicheventually converges to zero Finally Figure 10 shows the es-timated parameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ which as it can beseen converge towards the real values of the parameters of thesystem ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(66)

5 Conclusions

In this paper the adaptive control and the adaptive syn-chronization of a new financial system with unknownconstant parameters were studied in the cases of integer andfractional order e adaptive control law and the adaptivesynchronization law were designed based on Lyapunovrsquosstability theory and on the adaptive control theory e lawshave been designed in the cases of integer and incom-mensurate fractional order system

e proposed adaptive control technique is effective forchaos control and synchronization of the new financial

system when the constant parameters of the system areunknown Numerical simulations are carried out to provethe efficiency of the control and synchronization techniquesdesigned in this work

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank IMSP-UAC and the German AcademicExchange Service (DAAD) for financial support under theprogramme ldquoIn-CountryIn-Region Scholarship Programmerdquo

References

[1] K B Oldham and J Spanier 5e Fractional Calculus Aca-demic Press New York NY USA 1974

[2] B Ross ldquoFractional calculus and its applicationsrdquo LectureNotes in Mathematics Vol 457 Springer-Verlag New YorkNY USA 1975

7

6

5

4

3

2

1

0

ndash1

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

ρhat = 005 = ρ δhat = 1 = δlhat = 1 = l ehat = 03 = e

Figure 10 Parameter estimates for adaptive synchronization in fractional order case

14 Mathematical Problems in Engineering

[3] V Daftardar-Gejji and S Bhalekar ldquoChaos in fractional or-dered Liu systemrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1117ndash1127 2010

[4] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons amp Fractals vol 36no 5 pp 1305ndash1314 2008

[5] K Rajagopal A Bayani S Jafari A Karthikeyan andI Hussain ldquoChaotic dynamics of a fractional order glucose-insulin regulatory systemrdquo Frontiers of Information Tech-nology amp Electronic Engineering vol 21 no 7 pp 1108ndash11182019

[6] T T Hartley C F Lorenzo and H Killory Qammer ldquoChaosin a fractional order Chuarsquos systemrdquo IEEE Transactions onCircuits and Systems I Fundamental 5eory and Applicationsvol 42 no 8 pp 485ndash490 1995

[7] Q Jia ldquoChaos control and synchronization of the Newton-Leipnik chaotic systemrdquo Chaos Solitons amp Fractals vol 35no 4 pp 814ndash824 2008

[8] M T Yassen ldquoAdaptive control and synchronization of amodified Chuarsquos circuit systemrdquo Applied Mathematics andComputation vol 135 no 1 pp 113ndash128 2003

[9] V Sundarapandian ldquoAdaptive control and synchronizationof a generalized Lotka-Volterra systemrdquo International Journalof Bioinformatics amp Biosciences vol 1 no 1 pp 1ndash12 2011

[10] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A StatisticalMechanics and Its Applications vol 389 no 12 pp 2434ndash2442 2010

[11] U E Kocamaz A Goksu H Taskin and Y UyarogluldquoControl of chaotic two-predator one-prey model with singlestate control signalsrdquo Journal of Intelligent Manufacturingpp 1ndash10 2020

[12] A Hajipour and H Tavakoli ldquoDynamic analysis and adaptivesliding mode controller for a chaotic fractional incommen-surate order financial systemrdquo International Journal of Bi-furcation and Chaos vol 27 no 13 p 14 2017

[13] X Yi R Guo and Y Qi ldquoStabilization of chaotic systems withboth uncertainty and disturbance by the UDE-based controlmethodrdquo IEEE Access vol 8 no 1 pp 62471ndash62477 2020

[14] L Liu B Li and R Guo ldquoConsensus control for networkedmanipulators with switched parameters and topologiesrdquo IEEEAccess vol 9 pp 9209ndash9217 2021

[15] T Hou Y Liu and F Deng ldquoFinite horizon H2∕Hinfin controlfor SDEs with infinite Markovian jumpsrdquo Nonlinear AnalysisHybrid Systems vol 34 pp 108ndash120 2019

[16] R Xu and F Zhang ldquo-Nash mean-field games for generallinear-quadratic systems with applicationsrdquo Automaticavol 114 pp 1ndash4 2020

[17] R HilferApplications of Fractional Calculus in Physics WorldScientific Hackensack NJ USA 2001

[18] R He and P G Vaidya ldquoImplementation of chaotic cryp-tography with chaotic synchronizationrdquo Physical Review Evol 57 no 2 pp 1532ndash1535 1998

[19] J H Park ldquoChaos synchronization of a chaotic system vianonlinear controlrdquo Chaos Solitons amp Fractals vol 25 no 3pp 579ndash584 2005

[20] L Huang R Feng andMWang ldquoSynchronization of chaoticsystems via nonlinear controlrdquo Physics Letters A vol 320no 4 pp 271ndash275 2004

[21] S Bhalekar and V Daftardar-Gejji ldquoSynchronization ofdifferent fractional order chaotic systems using active con-trolrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3536ndash3546 2010

[22] H N Agiza and M T Yassen ldquoSynchronization of Rosslerand Chen chaotic dynamical systems using active controlrdquoPhysics Letters A vol 278 no 4 pp 191ndash197 2001

[23] T-l Liao ldquoAdaptive synchronization of two Lorenz SystemsrdquoChaos Solitons amp Fractals vol 9 no 9 pp 1555ndash1561 1998

[24] H Yu G Cai and Y Li ldquoDynamic analysis and control of anew hyperchaotic finance systemrdquo Nonlinear Dynamicsvol 67 no 3 pp 2171ndash2182 2012

[25] J-h Ma and Y-s Chen ldquoStudy for the bifurcation topologicalstructure and the global complicated character of a kind ofnonlinear finance system Irdquo Applied Mathematics and Me-chanics vol 22 no 11 pp 1240ndash1251 2001

[26] M S Abd-Elouahab N E Hamri and J Wang ldquoChaoscontrol of a fractional order financial systemrdquo MathematicalProblems in Engineering vol 2010 Article ID 27064618 pages 2010

[27] I Hernandez C Mateos J Nuntildeez and A F Tenorio ldquoLietheory applications to problems in mathematical finance andeconomicsrdquo Applied Mathematics and Computation vol 208no 2 pp 446ndash452 2009

[28] S A David J A T Machado D D Quintino andJ M Balthazar ldquoPartial chaos suppression in a fractionalorder macroeconomic modelrdquo Mathematics and Computersin Simulation vol 122 pp 55ndash68 2016

[29] J Yang E Zhang and M Liu ldquoBifurcation analysis and chaoscontrol in a modified finance system with delayed feedbackrdquoInternational Journal of Bifurcation and Chaos vol 26 no 062016

[30] A Hajipour and H Tavakoli ldquoAnalysis and circuit simulationof a novel nonlinear fractional incommensurate order fi-nancial systemrdquo Optik vol 127 no 22 pp 10643ndash106522016

[31] X Zhao Z Li and S Li ldquoSynchronization of a chaotic financesystemrdquo Applied Mathematics and Computation vol 217no 13 pp 6031ndash6039 2011

[32] Y Liao Y Zhou F Xu and X-B Shu ldquoA study on thecomplexity of a new chaotic financial systemrdquo Complexityvol 2020 Article ID 8821156 5 pages 2020

[33] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

[34] I Petras Fractional-order Nonlinear Systems ModelingAnalysis and Simulation Springer Science amp Business MediaBerlin Germany 2011

[35] K Diethelm N J Ford andA D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquo Nonlinear Dynamics vol 29 no 1ndash4 pp 3ndash222002

Mathematical Problems in Engineering 15

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

0 1 2 3 4 5 6 7 8 9 10Time

2

1

0

ndash1

2

1

0

ndash1

2

1

0

ndash1

x(t)

y(t)

z(t)

Figure 5 Time histories of the controlled fractional order new financial system (22)

7

8

6

5

4

3

2

1

0

ndash10 1 2 3 4 5 6 7 8 9 10

Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

δhat = 1 = δlhat = 01 = l ehat = 03 = e

ρhat = 005 = ρ

Para

met

er es

timat

es

Figure 6 Parameter estimates for adaptive control in fractional order case

Mathematical Problems in Engineering 9

By replacing the control law (39) in (38) we have

_e1 minus (e minus 1113954e)e1 +(d minus 1113954d)e3 minus h1e1

_e2 minus (k minus 1113954k) y22 minus y

211113872 1113873 minus (l minus 1113954l) x

22 minus x

211113872 1113873 minus h2e2

_e3 minus (c minus 1113954c)e3 minus (δ minus 1113954δ)e1 minus (ρ minus 1113954ρ)e2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(40)

Define the estimation error of unknown parameters asfollows

ed d minus 1113954d

ee e minus 1113954e

ek k minus 1113954k

el l minus 1113954l

em m minus 1113954m

ec c minus 1113954c

eδ δ minus 1113954δ

eρ ρ minus 1113954ρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(41)

By substituting equation (41) in system (40) system (40)becomes

_e1 minus eee1 + ede3 minus h1e1

_e2 minus ek y22 minus y

211113872 1113873 minus el x

22 minus x

211113872 1113873 minus h2e2

_e3 minus ece3 minus eδe1 minus eρe2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(42)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider the quadratic Lyapunov functiondefined as follows

V 12

e21 + e

22 + e

23 + e

2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(43)

which is a positive definite function on R11

e derivative with respect to time of equation (43) givesus

_V e1 _e1 + e2 _e2 + e3 _e3 + ed _ed + ee _ee

+ ek _ek + el _el + em _em + ec _ec + eδ _eδ + eρ _eρ(44)

Using system (42) and equation (15) equation (44)becomes

_V minus h1e21 minus h2e

22 minus h3e

23 + ee minus e

21 minus _1113954e1113872 1113873

+ed e1e3 minus_1113954d1113874 1113875 + ek minus e2 y

22 minus y

211113872 1113873 minus

_1113954k1113876 1113877

+el minus e2 x22 minus x

211113872 1113873 minus

_1113954l1113876 1113877 + ec minus e23 minus _1113954c1113872 1113873

+eδ minus e3e1 minus_1113954δ1113874 1113875 + eρ minus e3e2 minus _1113954ρ1113872 1113873 minus _1113954mem

(45)

From equation (45) we deduce that the estimated pa-rameters update law is

_1113954d e1e3 + h4ed_1113954e minus e

21 + h5ee

_1113954k minus e2 y22 minus y

211113872 1113873 + h6ek

_1113954l minus e2 x22 minus x

211113872 1113873 + h7el

_1113954m h8em_1113954c minus e

23 + h9ec

_1113954δ minus e3e1 + h10eδ_1113954ρ minus e3e2 + h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(46)

in which h4 h5 h6 h7 h8 h9 h10 and h11 are positiveconstants

By replacing equation (46) in equation(45) we get

_V minus h1e21 minus h2e

22 minus h3e

23 minus h4e

2d minus h5e

2e

minus h6e2k minus h7e

2l minus h8e

2m minus h9e

2c minus h10e

2δ minus h11e

2ρ lt 0

(47)

which is a negative definite function on R11 for positiveconstants hi (i 1 11) So we have found a functionwhich verifies the Lyapunov stability theorem (Vgt 0 _Vlt 0)

us the error functions converge globally and as-ymptotically towards zero for all initial conditions esynchronization of the states of the identical systems (35)and (36) is therefore complete So we have the followingresult

Theorem 3 5e identical financial systems (35) and (36)with unknown parameters are globally and asymptoticallysynchronized for all initial conditions by the adaptive controllaw (39) where the parameters update law is given by (46) andthe hi (i 1 11) are positive constants

412 Simulation Results In this part we use the fourth-order Rungendash-Kutta algorithm to solve the two identicalfinancial systems (35) and (36) with the adaptive control law(39) and the parameters update law (46) For the simulationthe time-step h 0001 is chosen e initial conditions forthe master system are (x1(0) y1(0) z1(0)) (12 15 16)

and for the slave system (x2(0) y2(0) z2(0))

(02 05 06)e parameters of the new financial system are chosen as

follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(48)

For the hi (i 1 11) of the adaptive and updatelaws we choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) minus 1 1113954k(0) 2 1113954c(0) 05

1113954m(0) 41113954l(0) 2 1113954ρ(0) 3

1113954d(0) 7 1113954δ(0) minus 05

(49)

10 Mathematical Problems in Engineering

8

4

0

ndash4

x 1(t)

x2(t)

x1(t)

x2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(a)

0 1 2 3 4 5 6 7 8 9 10Time t

3

2

1

0

y 1(t)

y2(t)

y1(t)

y2(t)

(b)

0 1 2 3 4 5 6 7 8 9 10Time t

24

0ndash2ndash4

z 1(t)

z2(t)

z1(t)

z2(t)

(c)

0 1 2 3 4 5 6 7 8 9 10Time t

1

0

ndash1

ndash2e 1(t)

e2(t)

e 3(t)

e1(t)

e2(t)e3(t)

(d)

Figure 7 (andashc) Time evolutions of the master and slave systems state variables (x1 x2) (y1 y2) (z1 z2) respectively in integer order caseand (d) time evolution of the error functions e1(black line) e2(red line) and e3(blue line)

7

6

5

4

3

2

1

0

ndash1

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

ρhat = 005 = ρ

δhat = 1 = δlhat = 01 = l ehat = 03 = e

khat = 002 = k

Figure 8 Parameter estimates for adaptive synchronization in integer order case

Mathematical Problems in Engineering 11

By applying the adaptive control law (39) and the pa-rameter update law (46) to the new controlled financialsystem (36) the results of the numerical simulations areshown in Figures 7(a)ndash7(d) and 8 From Figures 7(a)ndash7(c) itcan be seen that the state variables of the slave and mastersystems are synchronized Figure 7(d) shows the errorsystem which eventually converges to zero Finally Figure 8shows the estimated parameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρwhich as it can be seen converge towards the real values ofthe parameters of the system ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(50)

42FractionalOrderCase In this part the identical adaptivesynchronization of the new fractional order financial systemis achieved

421 Analytical Results Let us consider the master systemas being the system described with index 1 and the slavesystem as being the system described with index 2 We havetherefore for the master system the system

Dq1x1 dz1 + y1 minus e( 1113857x1

Dq2y1 minus ky

21 minus lx

21 + m

Dq3z1 minus cz1 minus δx1 minus ρy1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(51)

and for slave system we have

Dq1x2 dz2 + y2 minus e( 1113857x2 + u1

Dq2y2 minus ky

22 minus lx

22 + m + u2

Dq3z2 minus cz2 minus δx2 minus ρy2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(52)

where u1 u2 u3 are controllers to be designed so that system(52) synchronizes with system (51) and d e k l m

c δ and ρ are the unknown constant parameters of thesystem

Let us define the error functions between the statevariables of systems (2) and (51) as follows

e1 x2 minus x1

e2 y2 minus y1

e3 z2 minus z1

(53)

From equation (53) we get the following error system

Dq1e1 minus ee1 + de3 + y2x2 minus y1x1 + u1

Dq2e2 minus k y

22 minus y

211113872 1113873 minus l x

22 minus x

211113872 1113873 + u2

Dq3e3 minus ce3 minus δe1 minus ρe2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(54)

Let us define the adaptive control functionsui (i 1 2 3)

u1 1113954ee1 minus 1113954de3 minus y2x2 + y1x1 minus h1Dq1minus 1

e1 + v1

u1 1113954k y22 minus y

211113872 1113873 +1113954l x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

u3 1113954ce3 + 1113954δe1 + 1113954ρe2 minus h3Dq3minus 1

e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(55)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimates of theparameters d e k l m c δ and ρ respectively hi (i

1 2 3) are positive constants and vi (i 1 2 3) are non-linear functions that will be designed By replacing thecontrol law (55) in (54) we get

Dq1e1 minus (e minus 1113954e)e1 +(d minus 1113954d)e3 minus h1D

q1minus 1e1 + v1

Dq2e2 minus (k minus 1113954k) y

22 minus y

211113872 1113873 minus (l minus 1113954l) x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

Dq3e3 minus (c minus 1113954c)e3 minus (δ minus 1113954δ)e1 minus (ρ minus 1113954ρ)e2 minus h3D

q3minus 1e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(56)

With the estimation error of unknown parameters de-fined by equation (41) we obtain

Dq1e1 minus eee1 + ede3 minus h1D

q1minus 1e1 + v1

Dq2e2 minus ek y

22 minus y

211113872 1113873 minus el x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

Dq3e3 minus ece3 minus eδe1 minus eρe2 minus h3D

q3minus 1e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(57)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider the quadratic Lyapunov functiondefined as follows

V 12

e21 + e

22 + e

23 + e

2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(58)

which is a positive definite function on R11 e derivativewith respect to time of equation (58) gives us

_V e1 _e1 + e2 _e2 + e3 _e3 + ed _ed + ee _ee

+ ek _ek + el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

e1D1minus q1D

q1e1 + e2D1minus q2D

q2e2

+ e3D1minus q3D

q3e3 + ed _ed + ee _ee + ek _ek

+ el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

(59)

Taking into account system (57) and equation (15)equation (59) becomes

_V e1D1minus q1 minus eee1 + ede3 minus h1D

q1minus 1e1 + v11113960 1113961

+ e2D1minus q2 minus ek y

22 minus y

211113872 1113873 minus el x

22 minus x

211113872 11138731113960

minus h2Dq2minus 1

e2 + v21113961 + e3D1minus q3 minus ece3 minus eδe11113960

minus eρe2 minus h3Dq3minus 1

e3 + v31113961 minus_ded minus _eee minus

_kek

minus_lel minus _mem minus _1113954cec minus

_δeδ minus _ρeρ

(60)

From equation (60) we deduce that the estimated pa-rameters update law is

12 Mathematical Problems in Engineering

_1113954d h4ed

_1113954e h5ee

_1113954k h6ek

_1113954l h7el

_1113954m h8em

_1113954c h9ec

_1113954δ h10eδ

_1113954ρ h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

where h4 h5 h6 h7 h8 h9 h10 and h11 are positive constantsFrom equation (60) we also deduce that the nonlinear

functions vi (i 1 2 3) are given by

v1 eee1 minus ede3

v2 ek y22 minus y

211113872 1113873 + el x

22 minus x

211113872 1113873

v3 ece3 + eδe1 + eρe2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(62)

By replacing equations (61) and (62) in (60) we get

_V minus h1e21 minus h2e

22 minus h3e

23 minus h4e

2d minus h5e

2e

minus h6e2k minus h7e

2l minus h8e

2m minus h9e

2c minus h10e

2δ

minus h11e2ρ lt 0

(63)

which is a negative definite function on R11 for positiveconstants hi (i 1 11) So we have found a functionwhich verifies the Lyapunov stability theorem (Vgt 0 _Vlt 0)

us the error functions converge globally and as-ymptotically towards zero for all initial conditions esynchronization of the states of the identical systems (51)and (52) is therefore complete So we have the followingresult

Theorem 4 5e identical financial systems (51) and (52)with unknown parameters are globally and asymptoticallysynchronized for all initial conditions by the adaptive controllaw (55) with vi (i 1 2 3) given by (62) and where theparameters update law is given by (61) and thehi (i 1 11) are positive constants

422 Simulation Results In this part we use the Adams-Bashforth-Moulton predictor-corrector method proposedby Diethelm et al [35] to solve the two identical fractionalorder systems (51) and (52) with the adaptive control law(55) the vi (i 1 2 3) given by (62) and the law forupdating the parameters is given by (61) For the simulationthe time-step h 0001 is chosen e initial conditions forthe master system are (x1(0) y1(0) z1(0)) (12 15 16)

and for the slave system (x2(0) y2(0) z2(0))

(02 05 06) are usede orders qi (i 1 2 3) are taken asfollows (q1 q2 q3) (1 088 1) ie the case of incom-mensurate order

e parameters of the new fractional order financialsystem are chosen as follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(64)

0 1 2 3 4 5 6 7 8 9 10

8

4

0

ndash4

x 1(t)

x2(t)

x1(t)

x2(t)

Time t

(a)

3

2

1

0

y 1(t)

y2(t)

y1(t)

y2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(b)

20

ndash2ndash4

z 1(t)

z2(t)

z1(t)

z2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(c)

2

1

0

ndash1e 1(t)

e2(t)

e 3(t)

e1(t)

e2(t)e3(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(d)

Figure 9 (andashc) Time evolutions of the master and slave systems state variables (x1 x2) (y1 y2) (z1 z2) respectively in fractional ordercase and (d) time evolution of the error functions e1(black line) e2(red line) and e3(blue line)

Mathematical Problems in Engineering 13

For hi (i 1 11) of the adaptive and update lawswe choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) 1 1113954k(0) minus 1 1113954c(0) 5 1113954m(0) 61113954l(0) 3 1113954ρ(0) 4 1113954d(0) 3 1113954δ(0) 7 (65)

By applying the adaptive control law (55) and the parameterupdate law (61) to the new controlled fractional-order financialsystem (52) the results of numerical simulations are shown inFigures 9(a)ndash9(d) and 10 From Figures 9(a)ndash9(c) it can be seenthat the state variables of the master and slave systems aresynchronized Figure 9(d) shows the error system whicheventually converges to zero Finally Figure 10 shows the es-timated parameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ which as it can beseen converge towards the real values of the parameters of thesystem ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(66)

5 Conclusions

In this paper the adaptive control and the adaptive syn-chronization of a new financial system with unknownconstant parameters were studied in the cases of integer andfractional order e adaptive control law and the adaptivesynchronization law were designed based on Lyapunovrsquosstability theory and on the adaptive control theory e lawshave been designed in the cases of integer and incom-mensurate fractional order system

e proposed adaptive control technique is effective forchaos control and synchronization of the new financial

system when the constant parameters of the system areunknown Numerical simulations are carried out to provethe efficiency of the control and synchronization techniquesdesigned in this work

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank IMSP-UAC and the German AcademicExchange Service (DAAD) for financial support under theprogramme ldquoIn-CountryIn-Region Scholarship Programmerdquo

References

[1] K B Oldham and J Spanier 5e Fractional Calculus Aca-demic Press New York NY USA 1974

[2] B Ross ldquoFractional calculus and its applicationsrdquo LectureNotes in Mathematics Vol 457 Springer-Verlag New YorkNY USA 1975

7

6

5

4

3

2

1

0

ndash1

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

ρhat = 005 = ρ δhat = 1 = δlhat = 1 = l ehat = 03 = e

Figure 10 Parameter estimates for adaptive synchronization in fractional order case

14 Mathematical Problems in Engineering

[3] V Daftardar-Gejji and S Bhalekar ldquoChaos in fractional or-dered Liu systemrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1117ndash1127 2010

[4] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons amp Fractals vol 36no 5 pp 1305ndash1314 2008

[5] K Rajagopal A Bayani S Jafari A Karthikeyan andI Hussain ldquoChaotic dynamics of a fractional order glucose-insulin regulatory systemrdquo Frontiers of Information Tech-nology amp Electronic Engineering vol 21 no 7 pp 1108ndash11182019

[6] T T Hartley C F Lorenzo and H Killory Qammer ldquoChaosin a fractional order Chuarsquos systemrdquo IEEE Transactions onCircuits and Systems I Fundamental 5eory and Applicationsvol 42 no 8 pp 485ndash490 1995

[7] Q Jia ldquoChaos control and synchronization of the Newton-Leipnik chaotic systemrdquo Chaos Solitons amp Fractals vol 35no 4 pp 814ndash824 2008

[8] M T Yassen ldquoAdaptive control and synchronization of amodified Chuarsquos circuit systemrdquo Applied Mathematics andComputation vol 135 no 1 pp 113ndash128 2003

[9] V Sundarapandian ldquoAdaptive control and synchronizationof a generalized Lotka-Volterra systemrdquo International Journalof Bioinformatics amp Biosciences vol 1 no 1 pp 1ndash12 2011

[10] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A StatisticalMechanics and Its Applications vol 389 no 12 pp 2434ndash2442 2010

[11] U E Kocamaz A Goksu H Taskin and Y UyarogluldquoControl of chaotic two-predator one-prey model with singlestate control signalsrdquo Journal of Intelligent Manufacturingpp 1ndash10 2020

[12] A Hajipour and H Tavakoli ldquoDynamic analysis and adaptivesliding mode controller for a chaotic fractional incommen-surate order financial systemrdquo International Journal of Bi-furcation and Chaos vol 27 no 13 p 14 2017

[13] X Yi R Guo and Y Qi ldquoStabilization of chaotic systems withboth uncertainty and disturbance by the UDE-based controlmethodrdquo IEEE Access vol 8 no 1 pp 62471ndash62477 2020

[14] L Liu B Li and R Guo ldquoConsensus control for networkedmanipulators with switched parameters and topologiesrdquo IEEEAccess vol 9 pp 9209ndash9217 2021

[15] T Hou Y Liu and F Deng ldquoFinite horizon H2∕Hinfin controlfor SDEs with infinite Markovian jumpsrdquo Nonlinear AnalysisHybrid Systems vol 34 pp 108ndash120 2019

[16] R Xu and F Zhang ldquo-Nash mean-field games for generallinear-quadratic systems with applicationsrdquo Automaticavol 114 pp 1ndash4 2020

[17] R HilferApplications of Fractional Calculus in Physics WorldScientific Hackensack NJ USA 2001

[18] R He and P G Vaidya ldquoImplementation of chaotic cryp-tography with chaotic synchronizationrdquo Physical Review Evol 57 no 2 pp 1532ndash1535 1998

[19] J H Park ldquoChaos synchronization of a chaotic system vianonlinear controlrdquo Chaos Solitons amp Fractals vol 25 no 3pp 579ndash584 2005

[20] L Huang R Feng andMWang ldquoSynchronization of chaoticsystems via nonlinear controlrdquo Physics Letters A vol 320no 4 pp 271ndash275 2004

[21] S Bhalekar and V Daftardar-Gejji ldquoSynchronization ofdifferent fractional order chaotic systems using active con-trolrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3536ndash3546 2010

[22] H N Agiza and M T Yassen ldquoSynchronization of Rosslerand Chen chaotic dynamical systems using active controlrdquoPhysics Letters A vol 278 no 4 pp 191ndash197 2001

[23] T-l Liao ldquoAdaptive synchronization of two Lorenz SystemsrdquoChaos Solitons amp Fractals vol 9 no 9 pp 1555ndash1561 1998

[24] H Yu G Cai and Y Li ldquoDynamic analysis and control of anew hyperchaotic finance systemrdquo Nonlinear Dynamicsvol 67 no 3 pp 2171ndash2182 2012

[25] J-h Ma and Y-s Chen ldquoStudy for the bifurcation topologicalstructure and the global complicated character of a kind ofnonlinear finance system Irdquo Applied Mathematics and Me-chanics vol 22 no 11 pp 1240ndash1251 2001

[26] M S Abd-Elouahab N E Hamri and J Wang ldquoChaoscontrol of a fractional order financial systemrdquo MathematicalProblems in Engineering vol 2010 Article ID 27064618 pages 2010

[27] I Hernandez C Mateos J Nuntildeez and A F Tenorio ldquoLietheory applications to problems in mathematical finance andeconomicsrdquo Applied Mathematics and Computation vol 208no 2 pp 446ndash452 2009

[28] S A David J A T Machado D D Quintino andJ M Balthazar ldquoPartial chaos suppression in a fractionalorder macroeconomic modelrdquo Mathematics and Computersin Simulation vol 122 pp 55ndash68 2016

[29] J Yang E Zhang and M Liu ldquoBifurcation analysis and chaoscontrol in a modified finance system with delayed feedbackrdquoInternational Journal of Bifurcation and Chaos vol 26 no 062016

[30] A Hajipour and H Tavakoli ldquoAnalysis and circuit simulationof a novel nonlinear fractional incommensurate order fi-nancial systemrdquo Optik vol 127 no 22 pp 10643ndash106522016

[31] X Zhao Z Li and S Li ldquoSynchronization of a chaotic financesystemrdquo Applied Mathematics and Computation vol 217no 13 pp 6031ndash6039 2011

[32] Y Liao Y Zhou F Xu and X-B Shu ldquoA study on thecomplexity of a new chaotic financial systemrdquo Complexityvol 2020 Article ID 8821156 5 pages 2020

[33] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

[34] I Petras Fractional-order Nonlinear Systems ModelingAnalysis and Simulation Springer Science amp Business MediaBerlin Germany 2011

[35] K Diethelm N J Ford andA D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquo Nonlinear Dynamics vol 29 no 1ndash4 pp 3ndash222002

Mathematical Problems in Engineering 15

By replacing the control law (39) in (38) we have

_e1 minus (e minus 1113954e)e1 +(d minus 1113954d)e3 minus h1e1

_e2 minus (k minus 1113954k) y22 minus y

211113872 1113873 minus (l minus 1113954l) x

22 minus x

211113872 1113873 minus h2e2

_e3 minus (c minus 1113954c)e3 minus (δ minus 1113954δ)e1 minus (ρ minus 1113954ρ)e2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(40)

Define the estimation error of unknown parameters asfollows

ed d minus 1113954d

ee e minus 1113954e

ek k minus 1113954k

el l minus 1113954l

em m minus 1113954m

ec c minus 1113954c

eδ δ minus 1113954δ

eρ ρ minus 1113954ρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(41)

By substituting equation (41) in system (40) system (40)becomes

_e1 minus eee1 + ede3 minus h1e1

_e2 minus ek y22 minus y

211113872 1113873 minus el x

22 minus x

211113872 1113873 minus h2e2

_e3 minus ece3 minus eδe1 minus eρe2 minus h3e3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(42)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider the quadratic Lyapunov functiondefined as follows

V 12

e21 + e

22 + e

23 + e

2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(43)

which is a positive definite function on R11

e derivative with respect to time of equation (43) givesus

_V e1 _e1 + e2 _e2 + e3 _e3 + ed _ed + ee _ee

+ ek _ek + el _el + em _em + ec _ec + eδ _eδ + eρ _eρ(44)

Using system (42) and equation (15) equation (44)becomes

_V minus h1e21 minus h2e

22 minus h3e

23 + ee minus e

21 minus _1113954e1113872 1113873

+ed e1e3 minus_1113954d1113874 1113875 + ek minus e2 y

22 minus y

211113872 1113873 minus

_1113954k1113876 1113877

+el minus e2 x22 minus x

211113872 1113873 minus

_1113954l1113876 1113877 + ec minus e23 minus _1113954c1113872 1113873

+eδ minus e3e1 minus_1113954δ1113874 1113875 + eρ minus e3e2 minus _1113954ρ1113872 1113873 minus _1113954mem

(45)

From equation (45) we deduce that the estimated pa-rameters update law is

_1113954d e1e3 + h4ed_1113954e minus e

21 + h5ee

_1113954k minus e2 y22 minus y

211113872 1113873 + h6ek

_1113954l minus e2 x22 minus x

211113872 1113873 + h7el

_1113954m h8em_1113954c minus e

23 + h9ec

_1113954δ minus e3e1 + h10eδ_1113954ρ minus e3e2 + h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(46)

in which h4 h5 h6 h7 h8 h9 h10 and h11 are positiveconstants

By replacing equation (46) in equation(45) we get

_V minus h1e21 minus h2e

22 minus h3e

23 minus h4e

2d minus h5e

2e

minus h6e2k minus h7e

2l minus h8e

2m minus h9e

2c minus h10e

2δ minus h11e

2ρ lt 0

(47)

which is a negative definite function on R11 for positiveconstants hi (i 1 11) So we have found a functionwhich verifies the Lyapunov stability theorem (Vgt 0 _Vlt 0)

us the error functions converge globally and as-ymptotically towards zero for all initial conditions esynchronization of the states of the identical systems (35)and (36) is therefore complete So we have the followingresult

Theorem 3 5e identical financial systems (35) and (36)with unknown parameters are globally and asymptoticallysynchronized for all initial conditions by the adaptive controllaw (39) where the parameters update law is given by (46) andthe hi (i 1 11) are positive constants

412 Simulation Results In this part we use the fourth-order Rungendash-Kutta algorithm to solve the two identicalfinancial systems (35) and (36) with the adaptive control law(39) and the parameters update law (46) For the simulationthe time-step h 0001 is chosen e initial conditions forthe master system are (x1(0) y1(0) z1(0)) (12 15 16)

and for the slave system (x2(0) y2(0) z2(0))

(02 05 06)e parameters of the new financial system are chosen as

follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(48)

For the hi (i 1 11) of the adaptive and updatelaws we choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) minus 1 1113954k(0) 2 1113954c(0) 05

1113954m(0) 41113954l(0) 2 1113954ρ(0) 3

1113954d(0) 7 1113954δ(0) minus 05

(49)

10 Mathematical Problems in Engineering

8

4

0

ndash4

x 1(t)

x2(t)

x1(t)

x2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(a)

0 1 2 3 4 5 6 7 8 9 10Time t

3

2

1

0

y 1(t)

y2(t)

y1(t)

y2(t)

(b)

0 1 2 3 4 5 6 7 8 9 10Time t

24

0ndash2ndash4

z 1(t)

z2(t)

z1(t)

z2(t)

(c)

0 1 2 3 4 5 6 7 8 9 10Time t

1

0

ndash1

ndash2e 1(t)

e2(t)

e 3(t)

e1(t)

e2(t)e3(t)

(d)

Figure 7 (andashc) Time evolutions of the master and slave systems state variables (x1 x2) (y1 y2) (z1 z2) respectively in integer order caseand (d) time evolution of the error functions e1(black line) e2(red line) and e3(blue line)

7

6

5

4

3

2

1

0

ndash1

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

ρhat = 005 = ρ

δhat = 1 = δlhat = 01 = l ehat = 03 = e

khat = 002 = k

Figure 8 Parameter estimates for adaptive synchronization in integer order case

Mathematical Problems in Engineering 11

By applying the adaptive control law (39) and the pa-rameter update law (46) to the new controlled financialsystem (36) the results of the numerical simulations areshown in Figures 7(a)ndash7(d) and 8 From Figures 7(a)ndash7(c) itcan be seen that the state variables of the slave and mastersystems are synchronized Figure 7(d) shows the errorsystem which eventually converges to zero Finally Figure 8shows the estimated parameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρwhich as it can be seen converge towards the real values ofthe parameters of the system ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(50)

42FractionalOrderCase In this part the identical adaptivesynchronization of the new fractional order financial systemis achieved

421 Analytical Results Let us consider the master systemas being the system described with index 1 and the slavesystem as being the system described with index 2 We havetherefore for the master system the system

Dq1x1 dz1 + y1 minus e( 1113857x1

Dq2y1 minus ky

21 minus lx

21 + m

Dq3z1 minus cz1 minus δx1 minus ρy1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(51)

and for slave system we have

Dq1x2 dz2 + y2 minus e( 1113857x2 + u1

Dq2y2 minus ky

22 minus lx

22 + m + u2

Dq3z2 minus cz2 minus δx2 minus ρy2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(52)

where u1 u2 u3 are controllers to be designed so that system(52) synchronizes with system (51) and d e k l m

c δ and ρ are the unknown constant parameters of thesystem

Let us define the error functions between the statevariables of systems (2) and (51) as follows

e1 x2 minus x1

e2 y2 minus y1

e3 z2 minus z1

(53)

From equation (53) we get the following error system

Dq1e1 minus ee1 + de3 + y2x2 minus y1x1 + u1

Dq2e2 minus k y

22 minus y

211113872 1113873 minus l x

22 minus x

211113872 1113873 + u2

Dq3e3 minus ce3 minus δe1 minus ρe2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(54)

Let us define the adaptive control functionsui (i 1 2 3)

u1 1113954ee1 minus 1113954de3 minus y2x2 + y1x1 minus h1Dq1minus 1

e1 + v1

u1 1113954k y22 minus y

211113872 1113873 +1113954l x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

u3 1113954ce3 + 1113954δe1 + 1113954ρe2 minus h3Dq3minus 1

e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(55)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimates of theparameters d e k l m c δ and ρ respectively hi (i

1 2 3) are positive constants and vi (i 1 2 3) are non-linear functions that will be designed By replacing thecontrol law (55) in (54) we get

Dq1e1 minus (e minus 1113954e)e1 +(d minus 1113954d)e3 minus h1D

q1minus 1e1 + v1

Dq2e2 minus (k minus 1113954k) y

22 minus y

211113872 1113873 minus (l minus 1113954l) x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

Dq3e3 minus (c minus 1113954c)e3 minus (δ minus 1113954δ)e1 minus (ρ minus 1113954ρ)e2 minus h3D

q3minus 1e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(56)

With the estimation error of unknown parameters de-fined by equation (41) we obtain

Dq1e1 minus eee1 + ede3 minus h1D

q1minus 1e1 + v1

Dq2e2 minus ek y

22 minus y

211113872 1113873 minus el x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

Dq3e3 minus ece3 minus eδe1 minus eρe2 minus h3D

q3minus 1e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(57)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider the quadratic Lyapunov functiondefined as follows

V 12

e21 + e

22 + e

23 + e

2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(58)

which is a positive definite function on R11 e derivativewith respect to time of equation (58) gives us

_V e1 _e1 + e2 _e2 + e3 _e3 + ed _ed + ee _ee

+ ek _ek + el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

e1D1minus q1D

q1e1 + e2D1minus q2D

q2e2

+ e3D1minus q3D

q3e3 + ed _ed + ee _ee + ek _ek

+ el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

(59)

Taking into account system (57) and equation (15)equation (59) becomes

_V e1D1minus q1 minus eee1 + ede3 minus h1D

q1minus 1e1 + v11113960 1113961

+ e2D1minus q2 minus ek y

22 minus y

211113872 1113873 minus el x

22 minus x

211113872 11138731113960

minus h2Dq2minus 1

e2 + v21113961 + e3D1minus q3 minus ece3 minus eδe11113960

minus eρe2 minus h3Dq3minus 1

e3 + v31113961 minus_ded minus _eee minus

_kek

minus_lel minus _mem minus _1113954cec minus

_δeδ minus _ρeρ

(60)

From equation (60) we deduce that the estimated pa-rameters update law is

12 Mathematical Problems in Engineering

_1113954d h4ed

_1113954e h5ee

_1113954k h6ek

_1113954l h7el

_1113954m h8em

_1113954c h9ec

_1113954δ h10eδ

_1113954ρ h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

where h4 h5 h6 h7 h8 h9 h10 and h11 are positive constantsFrom equation (60) we also deduce that the nonlinear

functions vi (i 1 2 3) are given by

v1 eee1 minus ede3

v2 ek y22 minus y

211113872 1113873 + el x

22 minus x

211113872 1113873

v3 ece3 + eδe1 + eρe2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(62)

By replacing equations (61) and (62) in (60) we get

_V minus h1e21 minus h2e

22 minus h3e

23 minus h4e

2d minus h5e

2e

minus h6e2k minus h7e

2l minus h8e

2m minus h9e

2c minus h10e

2δ

minus h11e2ρ lt 0

(63)

which is a negative definite function on R11 for positiveconstants hi (i 1 11) So we have found a functionwhich verifies the Lyapunov stability theorem (Vgt 0 _Vlt 0)

us the error functions converge globally and as-ymptotically towards zero for all initial conditions esynchronization of the states of the identical systems (51)and (52) is therefore complete So we have the followingresult

Theorem 4 5e identical financial systems (51) and (52)with unknown parameters are globally and asymptoticallysynchronized for all initial conditions by the adaptive controllaw (55) with vi (i 1 2 3) given by (62) and where theparameters update law is given by (61) and thehi (i 1 11) are positive constants

422 Simulation Results In this part we use the Adams-Bashforth-Moulton predictor-corrector method proposedby Diethelm et al [35] to solve the two identical fractionalorder systems (51) and (52) with the adaptive control law(55) the vi (i 1 2 3) given by (62) and the law forupdating the parameters is given by (61) For the simulationthe time-step h 0001 is chosen e initial conditions forthe master system are (x1(0) y1(0) z1(0)) (12 15 16)

and for the slave system (x2(0) y2(0) z2(0))

(02 05 06) are usede orders qi (i 1 2 3) are taken asfollows (q1 q2 q3) (1 088 1) ie the case of incom-mensurate order

e parameters of the new fractional order financialsystem are chosen as follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(64)

0 1 2 3 4 5 6 7 8 9 10

8

4

0

ndash4

x 1(t)

x2(t)

x1(t)

x2(t)

Time t

(a)

3

2

1

0

y 1(t)

y2(t)

y1(t)

y2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(b)

20

ndash2ndash4

z 1(t)

z2(t)

z1(t)

z2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(c)

2

1

0

ndash1e 1(t)

e2(t)

e 3(t)

e1(t)

e2(t)e3(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(d)

Figure 9 (andashc) Time evolutions of the master and slave systems state variables (x1 x2) (y1 y2) (z1 z2) respectively in fractional ordercase and (d) time evolution of the error functions e1(black line) e2(red line) and e3(blue line)

Mathematical Problems in Engineering 13

For hi (i 1 11) of the adaptive and update lawswe choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) 1 1113954k(0) minus 1 1113954c(0) 5 1113954m(0) 61113954l(0) 3 1113954ρ(0) 4 1113954d(0) 3 1113954δ(0) 7 (65)

By applying the adaptive control law (55) and the parameterupdate law (61) to the new controlled fractional-order financialsystem (52) the results of numerical simulations are shown inFigures 9(a)ndash9(d) and 10 From Figures 9(a)ndash9(c) it can be seenthat the state variables of the master and slave systems aresynchronized Figure 9(d) shows the error system whicheventually converges to zero Finally Figure 10 shows the es-timated parameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ which as it can beseen converge towards the real values of the parameters of thesystem ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(66)

5 Conclusions

In this paper the adaptive control and the adaptive syn-chronization of a new financial system with unknownconstant parameters were studied in the cases of integer andfractional order e adaptive control law and the adaptivesynchronization law were designed based on Lyapunovrsquosstability theory and on the adaptive control theory e lawshave been designed in the cases of integer and incom-mensurate fractional order system

e proposed adaptive control technique is effective forchaos control and synchronization of the new financial

system when the constant parameters of the system areunknown Numerical simulations are carried out to provethe efficiency of the control and synchronization techniquesdesigned in this work

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank IMSP-UAC and the German AcademicExchange Service (DAAD) for financial support under theprogramme ldquoIn-CountryIn-Region Scholarship Programmerdquo

References

[1] K B Oldham and J Spanier 5e Fractional Calculus Aca-demic Press New York NY USA 1974

[2] B Ross ldquoFractional calculus and its applicationsrdquo LectureNotes in Mathematics Vol 457 Springer-Verlag New YorkNY USA 1975

7

6

5

4

3

2

1

0

ndash1

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

ρhat = 005 = ρ δhat = 1 = δlhat = 1 = l ehat = 03 = e

Figure 10 Parameter estimates for adaptive synchronization in fractional order case

14 Mathematical Problems in Engineering

[3] V Daftardar-Gejji and S Bhalekar ldquoChaos in fractional or-dered Liu systemrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1117ndash1127 2010

[4] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons amp Fractals vol 36no 5 pp 1305ndash1314 2008

[5] K Rajagopal A Bayani S Jafari A Karthikeyan andI Hussain ldquoChaotic dynamics of a fractional order glucose-insulin regulatory systemrdquo Frontiers of Information Tech-nology amp Electronic Engineering vol 21 no 7 pp 1108ndash11182019

[6] T T Hartley C F Lorenzo and H Killory Qammer ldquoChaosin a fractional order Chuarsquos systemrdquo IEEE Transactions onCircuits and Systems I Fundamental 5eory and Applicationsvol 42 no 8 pp 485ndash490 1995

[7] Q Jia ldquoChaos control and synchronization of the Newton-Leipnik chaotic systemrdquo Chaos Solitons amp Fractals vol 35no 4 pp 814ndash824 2008

[8] M T Yassen ldquoAdaptive control and synchronization of amodified Chuarsquos circuit systemrdquo Applied Mathematics andComputation vol 135 no 1 pp 113ndash128 2003

[9] V Sundarapandian ldquoAdaptive control and synchronizationof a generalized Lotka-Volterra systemrdquo International Journalof Bioinformatics amp Biosciences vol 1 no 1 pp 1ndash12 2011

[10] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A StatisticalMechanics and Its Applications vol 389 no 12 pp 2434ndash2442 2010

[11] U E Kocamaz A Goksu H Taskin and Y UyarogluldquoControl of chaotic two-predator one-prey model with singlestate control signalsrdquo Journal of Intelligent Manufacturingpp 1ndash10 2020

[12] A Hajipour and H Tavakoli ldquoDynamic analysis and adaptivesliding mode controller for a chaotic fractional incommen-surate order financial systemrdquo International Journal of Bi-furcation and Chaos vol 27 no 13 p 14 2017

[13] X Yi R Guo and Y Qi ldquoStabilization of chaotic systems withboth uncertainty and disturbance by the UDE-based controlmethodrdquo IEEE Access vol 8 no 1 pp 62471ndash62477 2020

[14] L Liu B Li and R Guo ldquoConsensus control for networkedmanipulators with switched parameters and topologiesrdquo IEEEAccess vol 9 pp 9209ndash9217 2021

[15] T Hou Y Liu and F Deng ldquoFinite horizon H2∕Hinfin controlfor SDEs with infinite Markovian jumpsrdquo Nonlinear AnalysisHybrid Systems vol 34 pp 108ndash120 2019

[16] R Xu and F Zhang ldquo-Nash mean-field games for generallinear-quadratic systems with applicationsrdquo Automaticavol 114 pp 1ndash4 2020

[17] R HilferApplications of Fractional Calculus in Physics WorldScientific Hackensack NJ USA 2001

[18] R He and P G Vaidya ldquoImplementation of chaotic cryp-tography with chaotic synchronizationrdquo Physical Review Evol 57 no 2 pp 1532ndash1535 1998

[19] J H Park ldquoChaos synchronization of a chaotic system vianonlinear controlrdquo Chaos Solitons amp Fractals vol 25 no 3pp 579ndash584 2005

[20] L Huang R Feng andMWang ldquoSynchronization of chaoticsystems via nonlinear controlrdquo Physics Letters A vol 320no 4 pp 271ndash275 2004

[21] S Bhalekar and V Daftardar-Gejji ldquoSynchronization ofdifferent fractional order chaotic systems using active con-trolrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3536ndash3546 2010

[22] H N Agiza and M T Yassen ldquoSynchronization of Rosslerand Chen chaotic dynamical systems using active controlrdquoPhysics Letters A vol 278 no 4 pp 191ndash197 2001

[23] T-l Liao ldquoAdaptive synchronization of two Lorenz SystemsrdquoChaos Solitons amp Fractals vol 9 no 9 pp 1555ndash1561 1998

[24] H Yu G Cai and Y Li ldquoDynamic analysis and control of anew hyperchaotic finance systemrdquo Nonlinear Dynamicsvol 67 no 3 pp 2171ndash2182 2012

[25] J-h Ma and Y-s Chen ldquoStudy for the bifurcation topologicalstructure and the global complicated character of a kind ofnonlinear finance system Irdquo Applied Mathematics and Me-chanics vol 22 no 11 pp 1240ndash1251 2001

[26] M S Abd-Elouahab N E Hamri and J Wang ldquoChaoscontrol of a fractional order financial systemrdquo MathematicalProblems in Engineering vol 2010 Article ID 27064618 pages 2010

[27] I Hernandez C Mateos J Nuntildeez and A F Tenorio ldquoLietheory applications to problems in mathematical finance andeconomicsrdquo Applied Mathematics and Computation vol 208no 2 pp 446ndash452 2009

[28] S A David J A T Machado D D Quintino andJ M Balthazar ldquoPartial chaos suppression in a fractionalorder macroeconomic modelrdquo Mathematics and Computersin Simulation vol 122 pp 55ndash68 2016

[29] J Yang E Zhang and M Liu ldquoBifurcation analysis and chaoscontrol in a modified finance system with delayed feedbackrdquoInternational Journal of Bifurcation and Chaos vol 26 no 062016

[30] A Hajipour and H Tavakoli ldquoAnalysis and circuit simulationof a novel nonlinear fractional incommensurate order fi-nancial systemrdquo Optik vol 127 no 22 pp 10643ndash106522016

[31] X Zhao Z Li and S Li ldquoSynchronization of a chaotic financesystemrdquo Applied Mathematics and Computation vol 217no 13 pp 6031ndash6039 2011

[32] Y Liao Y Zhou F Xu and X-B Shu ldquoA study on thecomplexity of a new chaotic financial systemrdquo Complexityvol 2020 Article ID 8821156 5 pages 2020

[33] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

[34] I Petras Fractional-order Nonlinear Systems ModelingAnalysis and Simulation Springer Science amp Business MediaBerlin Germany 2011

[35] K Diethelm N J Ford andA D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquo Nonlinear Dynamics vol 29 no 1ndash4 pp 3ndash222002

Mathematical Problems in Engineering 15

8

4

0

ndash4

x 1(t)

x2(t)

x1(t)

x2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(a)

0 1 2 3 4 5 6 7 8 9 10Time t

3

2

1

0

y 1(t)

y2(t)

y1(t)

y2(t)

(b)

0 1 2 3 4 5 6 7 8 9 10Time t

24

0ndash2ndash4

z 1(t)

z2(t)

z1(t)

z2(t)

(c)

0 1 2 3 4 5 6 7 8 9 10Time t

1

0

ndash1

ndash2e 1(t)

e2(t)

e 3(t)

e1(t)

e2(t)e3(t)

(d)

Figure 7 (andashc) Time evolutions of the master and slave systems state variables (x1 x2) (y1 y2) (z1 z2) respectively in integer order caseand (d) time evolution of the error functions e1(black line) e2(red line) and e3(blue line)

7

6

5

4

3

2

1

0

ndash1

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

ρhat = 005 = ρ

δhat = 1 = δlhat = 01 = l ehat = 03 = e

khat = 002 = k

Figure 8 Parameter estimates for adaptive synchronization in integer order case

Mathematical Problems in Engineering 11

By applying the adaptive control law (39) and the pa-rameter update law (46) to the new controlled financialsystem (36) the results of the numerical simulations areshown in Figures 7(a)ndash7(d) and 8 From Figures 7(a)ndash7(c) itcan be seen that the state variables of the slave and mastersystems are synchronized Figure 7(d) shows the errorsystem which eventually converges to zero Finally Figure 8shows the estimated parameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρwhich as it can be seen converge towards the real values ofthe parameters of the system ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(50)

42FractionalOrderCase In this part the identical adaptivesynchronization of the new fractional order financial systemis achieved

421 Analytical Results Let us consider the master systemas being the system described with index 1 and the slavesystem as being the system described with index 2 We havetherefore for the master system the system

Dq1x1 dz1 + y1 minus e( 1113857x1

Dq2y1 minus ky

21 minus lx

21 + m

Dq3z1 minus cz1 minus δx1 minus ρy1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(51)

and for slave system we have

Dq1x2 dz2 + y2 minus e( 1113857x2 + u1

Dq2y2 minus ky

22 minus lx

22 + m + u2

Dq3z2 minus cz2 minus δx2 minus ρy2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(52)

where u1 u2 u3 are controllers to be designed so that system(52) synchronizes with system (51) and d e k l m

c δ and ρ are the unknown constant parameters of thesystem

Let us define the error functions between the statevariables of systems (2) and (51) as follows

e1 x2 minus x1

e2 y2 minus y1

e3 z2 minus z1

(53)

From equation (53) we get the following error system

Dq1e1 minus ee1 + de3 + y2x2 minus y1x1 + u1

Dq2e2 minus k y

22 minus y

211113872 1113873 minus l x

22 minus x

211113872 1113873 + u2

Dq3e3 minus ce3 minus δe1 minus ρe2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(54)

Let us define the adaptive control functionsui (i 1 2 3)

u1 1113954ee1 minus 1113954de3 minus y2x2 + y1x1 minus h1Dq1minus 1

e1 + v1

u1 1113954k y22 minus y

211113872 1113873 +1113954l x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

u3 1113954ce3 + 1113954δe1 + 1113954ρe2 minus h3Dq3minus 1

e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(55)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimates of theparameters d e k l m c δ and ρ respectively hi (i

1 2 3) are positive constants and vi (i 1 2 3) are non-linear functions that will be designed By replacing thecontrol law (55) in (54) we get

Dq1e1 minus (e minus 1113954e)e1 +(d minus 1113954d)e3 minus h1D

q1minus 1e1 + v1

Dq2e2 minus (k minus 1113954k) y

22 minus y

211113872 1113873 minus (l minus 1113954l) x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

Dq3e3 minus (c minus 1113954c)e3 minus (δ minus 1113954δ)e1 minus (ρ minus 1113954ρ)e2 minus h3D

q3minus 1e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(56)

With the estimation error of unknown parameters de-fined by equation (41) we obtain

Dq1e1 minus eee1 + ede3 minus h1D

q1minus 1e1 + v1

Dq2e2 minus ek y

22 minus y

211113872 1113873 minus el x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

Dq3e3 minus ece3 minus eδe1 minus eρe2 minus h3D

q3minus 1e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(57)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider the quadratic Lyapunov functiondefined as follows

V 12

e21 + e

22 + e

23 + e

2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(58)

which is a positive definite function on R11 e derivativewith respect to time of equation (58) gives us

_V e1 _e1 + e2 _e2 + e3 _e3 + ed _ed + ee _ee

+ ek _ek + el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

e1D1minus q1D

q1e1 + e2D1minus q2D

q2e2

+ e3D1minus q3D

q3e3 + ed _ed + ee _ee + ek _ek

+ el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

(59)

Taking into account system (57) and equation (15)equation (59) becomes

_V e1D1minus q1 minus eee1 + ede3 minus h1D

q1minus 1e1 + v11113960 1113961

+ e2D1minus q2 minus ek y

22 minus y

211113872 1113873 minus el x

22 minus x

211113872 11138731113960

minus h2Dq2minus 1

e2 + v21113961 + e3D1minus q3 minus ece3 minus eδe11113960

minus eρe2 minus h3Dq3minus 1

e3 + v31113961 minus_ded minus _eee minus

_kek

minus_lel minus _mem minus _1113954cec minus

_δeδ minus _ρeρ

(60)

From equation (60) we deduce that the estimated pa-rameters update law is

12 Mathematical Problems in Engineering

_1113954d h4ed

_1113954e h5ee

_1113954k h6ek

_1113954l h7el

_1113954m h8em

_1113954c h9ec

_1113954δ h10eδ

_1113954ρ h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

where h4 h5 h6 h7 h8 h9 h10 and h11 are positive constantsFrom equation (60) we also deduce that the nonlinear

functions vi (i 1 2 3) are given by

v1 eee1 minus ede3

v2 ek y22 minus y

211113872 1113873 + el x

22 minus x

211113872 1113873

v3 ece3 + eδe1 + eρe2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(62)

By replacing equations (61) and (62) in (60) we get

_V minus h1e21 minus h2e

22 minus h3e

23 minus h4e

2d minus h5e

2e

minus h6e2k minus h7e

2l minus h8e

2m minus h9e

2c minus h10e

2δ

minus h11e2ρ lt 0

(63)

which is a negative definite function on R11 for positiveconstants hi (i 1 11) So we have found a functionwhich verifies the Lyapunov stability theorem (Vgt 0 _Vlt 0)

us the error functions converge globally and as-ymptotically towards zero for all initial conditions esynchronization of the states of the identical systems (51)and (52) is therefore complete So we have the followingresult

Theorem 4 5e identical financial systems (51) and (52)with unknown parameters are globally and asymptoticallysynchronized for all initial conditions by the adaptive controllaw (55) with vi (i 1 2 3) given by (62) and where theparameters update law is given by (61) and thehi (i 1 11) are positive constants

422 Simulation Results In this part we use the Adams-Bashforth-Moulton predictor-corrector method proposedby Diethelm et al [35] to solve the two identical fractionalorder systems (51) and (52) with the adaptive control law(55) the vi (i 1 2 3) given by (62) and the law forupdating the parameters is given by (61) For the simulationthe time-step h 0001 is chosen e initial conditions forthe master system are (x1(0) y1(0) z1(0)) (12 15 16)

and for the slave system (x2(0) y2(0) z2(0))

(02 05 06) are usede orders qi (i 1 2 3) are taken asfollows (q1 q2 q3) (1 088 1) ie the case of incom-mensurate order

e parameters of the new fractional order financialsystem are chosen as follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(64)

0 1 2 3 4 5 6 7 8 9 10

8

4

0

ndash4

x 1(t)

x2(t)

x1(t)

x2(t)

Time t

(a)

3

2

1

0

y 1(t)

y2(t)

y1(t)

y2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(b)

20

ndash2ndash4

z 1(t)

z2(t)

z1(t)

z2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(c)

2

1

0

ndash1e 1(t)

e2(t)

e 3(t)

e1(t)

e2(t)e3(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(d)

Figure 9 (andashc) Time evolutions of the master and slave systems state variables (x1 x2) (y1 y2) (z1 z2) respectively in fractional ordercase and (d) time evolution of the error functions e1(black line) e2(red line) and e3(blue line)

Mathematical Problems in Engineering 13

For hi (i 1 11) of the adaptive and update lawswe choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) 1 1113954k(0) minus 1 1113954c(0) 5 1113954m(0) 61113954l(0) 3 1113954ρ(0) 4 1113954d(0) 3 1113954δ(0) 7 (65)

By applying the adaptive control law (55) and the parameterupdate law (61) to the new controlled fractional-order financialsystem (52) the results of numerical simulations are shown inFigures 9(a)ndash9(d) and 10 From Figures 9(a)ndash9(c) it can be seenthat the state variables of the master and slave systems aresynchronized Figure 9(d) shows the error system whicheventually converges to zero Finally Figure 10 shows the es-timated parameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ which as it can beseen converge towards the real values of the parameters of thesystem ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(66)

5 Conclusions

In this paper the adaptive control and the adaptive syn-chronization of a new financial system with unknownconstant parameters were studied in the cases of integer andfractional order e adaptive control law and the adaptivesynchronization law were designed based on Lyapunovrsquosstability theory and on the adaptive control theory e lawshave been designed in the cases of integer and incom-mensurate fractional order system

e proposed adaptive control technique is effective forchaos control and synchronization of the new financial

system when the constant parameters of the system areunknown Numerical simulations are carried out to provethe efficiency of the control and synchronization techniquesdesigned in this work

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank IMSP-UAC and the German AcademicExchange Service (DAAD) for financial support under theprogramme ldquoIn-CountryIn-Region Scholarship Programmerdquo

References

[1] K B Oldham and J Spanier 5e Fractional Calculus Aca-demic Press New York NY USA 1974

[2] B Ross ldquoFractional calculus and its applicationsrdquo LectureNotes in Mathematics Vol 457 Springer-Verlag New YorkNY USA 1975

7

6

5

4

3

2

1

0

ndash1

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

ρhat = 005 = ρ δhat = 1 = δlhat = 1 = l ehat = 03 = e

Figure 10 Parameter estimates for adaptive synchronization in fractional order case

14 Mathematical Problems in Engineering

[3] V Daftardar-Gejji and S Bhalekar ldquoChaos in fractional or-dered Liu systemrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1117ndash1127 2010

[4] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons amp Fractals vol 36no 5 pp 1305ndash1314 2008

[5] K Rajagopal A Bayani S Jafari A Karthikeyan andI Hussain ldquoChaotic dynamics of a fractional order glucose-insulin regulatory systemrdquo Frontiers of Information Tech-nology amp Electronic Engineering vol 21 no 7 pp 1108ndash11182019

[6] T T Hartley C F Lorenzo and H Killory Qammer ldquoChaosin a fractional order Chuarsquos systemrdquo IEEE Transactions onCircuits and Systems I Fundamental 5eory and Applicationsvol 42 no 8 pp 485ndash490 1995

[7] Q Jia ldquoChaos control and synchronization of the Newton-Leipnik chaotic systemrdquo Chaos Solitons amp Fractals vol 35no 4 pp 814ndash824 2008

[8] M T Yassen ldquoAdaptive control and synchronization of amodified Chuarsquos circuit systemrdquo Applied Mathematics andComputation vol 135 no 1 pp 113ndash128 2003

[9] V Sundarapandian ldquoAdaptive control and synchronizationof a generalized Lotka-Volterra systemrdquo International Journalof Bioinformatics amp Biosciences vol 1 no 1 pp 1ndash12 2011

[10] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A StatisticalMechanics and Its Applications vol 389 no 12 pp 2434ndash2442 2010

[11] U E Kocamaz A Goksu H Taskin and Y UyarogluldquoControl of chaotic two-predator one-prey model with singlestate control signalsrdquo Journal of Intelligent Manufacturingpp 1ndash10 2020

[12] A Hajipour and H Tavakoli ldquoDynamic analysis and adaptivesliding mode controller for a chaotic fractional incommen-surate order financial systemrdquo International Journal of Bi-furcation and Chaos vol 27 no 13 p 14 2017

[13] X Yi R Guo and Y Qi ldquoStabilization of chaotic systems withboth uncertainty and disturbance by the UDE-based controlmethodrdquo IEEE Access vol 8 no 1 pp 62471ndash62477 2020

[14] L Liu B Li and R Guo ldquoConsensus control for networkedmanipulators with switched parameters and topologiesrdquo IEEEAccess vol 9 pp 9209ndash9217 2021

[15] T Hou Y Liu and F Deng ldquoFinite horizon H2∕Hinfin controlfor SDEs with infinite Markovian jumpsrdquo Nonlinear AnalysisHybrid Systems vol 34 pp 108ndash120 2019

[16] R Xu and F Zhang ldquo-Nash mean-field games for generallinear-quadratic systems with applicationsrdquo Automaticavol 114 pp 1ndash4 2020

[17] R HilferApplications of Fractional Calculus in Physics WorldScientific Hackensack NJ USA 2001

[18] R He and P G Vaidya ldquoImplementation of chaotic cryp-tography with chaotic synchronizationrdquo Physical Review Evol 57 no 2 pp 1532ndash1535 1998

[19] J H Park ldquoChaos synchronization of a chaotic system vianonlinear controlrdquo Chaos Solitons amp Fractals vol 25 no 3pp 579ndash584 2005

[20] L Huang R Feng andMWang ldquoSynchronization of chaoticsystems via nonlinear controlrdquo Physics Letters A vol 320no 4 pp 271ndash275 2004

[21] S Bhalekar and V Daftardar-Gejji ldquoSynchronization ofdifferent fractional order chaotic systems using active con-trolrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3536ndash3546 2010

[22] H N Agiza and M T Yassen ldquoSynchronization of Rosslerand Chen chaotic dynamical systems using active controlrdquoPhysics Letters A vol 278 no 4 pp 191ndash197 2001

[23] T-l Liao ldquoAdaptive synchronization of two Lorenz SystemsrdquoChaos Solitons amp Fractals vol 9 no 9 pp 1555ndash1561 1998

[24] H Yu G Cai and Y Li ldquoDynamic analysis and control of anew hyperchaotic finance systemrdquo Nonlinear Dynamicsvol 67 no 3 pp 2171ndash2182 2012

[25] J-h Ma and Y-s Chen ldquoStudy for the bifurcation topologicalstructure and the global complicated character of a kind ofnonlinear finance system Irdquo Applied Mathematics and Me-chanics vol 22 no 11 pp 1240ndash1251 2001

[26] M S Abd-Elouahab N E Hamri and J Wang ldquoChaoscontrol of a fractional order financial systemrdquo MathematicalProblems in Engineering vol 2010 Article ID 27064618 pages 2010

[27] I Hernandez C Mateos J Nuntildeez and A F Tenorio ldquoLietheory applications to problems in mathematical finance andeconomicsrdquo Applied Mathematics and Computation vol 208no 2 pp 446ndash452 2009

[28] S A David J A T Machado D D Quintino andJ M Balthazar ldquoPartial chaos suppression in a fractionalorder macroeconomic modelrdquo Mathematics and Computersin Simulation vol 122 pp 55ndash68 2016

[29] J Yang E Zhang and M Liu ldquoBifurcation analysis and chaoscontrol in a modified finance system with delayed feedbackrdquoInternational Journal of Bifurcation and Chaos vol 26 no 062016

[30] A Hajipour and H Tavakoli ldquoAnalysis and circuit simulationof a novel nonlinear fractional incommensurate order fi-nancial systemrdquo Optik vol 127 no 22 pp 10643ndash106522016

[31] X Zhao Z Li and S Li ldquoSynchronization of a chaotic financesystemrdquo Applied Mathematics and Computation vol 217no 13 pp 6031ndash6039 2011

[32] Y Liao Y Zhou F Xu and X-B Shu ldquoA study on thecomplexity of a new chaotic financial systemrdquo Complexityvol 2020 Article ID 8821156 5 pages 2020

[33] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

[34] I Petras Fractional-order Nonlinear Systems ModelingAnalysis and Simulation Springer Science amp Business MediaBerlin Germany 2011

[35] K Diethelm N J Ford andA D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquo Nonlinear Dynamics vol 29 no 1ndash4 pp 3ndash222002

Mathematical Problems in Engineering 15

By applying the adaptive control law (39) and the pa-rameter update law (46) to the new controlled financialsystem (36) the results of the numerical simulations areshown in Figures 7(a)ndash7(d) and 8 From Figures 7(a)ndash7(c) itcan be seen that the state variables of the slave and mastersystems are synchronized Figure 7(d) shows the errorsystem which eventually converges to zero Finally Figure 8shows the estimated parameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρwhich as it can be seen converge towards the real values ofthe parameters of the system ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(50)

42FractionalOrderCase In this part the identical adaptivesynchronization of the new fractional order financial systemis achieved

421 Analytical Results Let us consider the master systemas being the system described with index 1 and the slavesystem as being the system described with index 2 We havetherefore for the master system the system

Dq1x1 dz1 + y1 minus e( 1113857x1

Dq2y1 minus ky

21 minus lx

21 + m

Dq3z1 minus cz1 minus δx1 minus ρy1

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(51)

and for slave system we have

Dq1x2 dz2 + y2 minus e( 1113857x2 + u1

Dq2y2 minus ky

22 minus lx

22 + m + u2

Dq3z2 minus cz2 minus δx2 minus ρy2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(52)

where u1 u2 u3 are controllers to be designed so that system(52) synchronizes with system (51) and d e k l m

c δ and ρ are the unknown constant parameters of thesystem

Let us define the error functions between the statevariables of systems (2) and (51) as follows

e1 x2 minus x1

e2 y2 minus y1

e3 z2 minus z1

(53)

From equation (53) we get the following error system

Dq1e1 minus ee1 + de3 + y2x2 minus y1x1 + u1

Dq2e2 minus k y

22 minus y

211113872 1113873 minus l x

22 minus x

211113872 1113873 + u2

Dq3e3 minus ce3 minus δe1 minus ρe2 + u3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(54)

Let us define the adaptive control functionsui (i 1 2 3)

u1 1113954ee1 minus 1113954de3 minus y2x2 + y1x1 minus h1Dq1minus 1

e1 + v1

u1 1113954k y22 minus y

211113872 1113873 +1113954l x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

u3 1113954ce3 + 1113954δe1 + 1113954ρe2 minus h3Dq3minus 1

e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(55)

in which 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ are the estimates of theparameters d e k l m c δ and ρ respectively hi (i

1 2 3) are positive constants and vi (i 1 2 3) are non-linear functions that will be designed By replacing thecontrol law (55) in (54) we get

Dq1e1 minus (e minus 1113954e)e1 +(d minus 1113954d)e3 minus h1D

q1minus 1e1 + v1

Dq2e2 minus (k minus 1113954k) y

22 minus y

211113872 1113873 minus (l minus 1113954l) x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

Dq3e3 minus (c minus 1113954c)e3 minus (δ minus 1113954δ)e1 minus (ρ minus 1113954ρ)e2 minus h3D

q3minus 1e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(56)

With the estimation error of unknown parameters de-fined by equation (41) we obtain

Dq1e1 minus eee1 + ede3 minus h1D

q1minus 1e1 + v1

Dq2e2 minus ek y

22 minus y

211113872 1113873 minus el x

22 minus x

211113872 1113873 minus h2D

q2minus 1e2 + v2

Dq3e3 minus ece3 minus eδe1 minus eρe2 minus h3D

q3minus 1e3 + v3

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(57)

For the design of the parameter update law which willallow to adjusting the parameter estimates we use Lyapu-novrsquos stability theory

For this consider the quadratic Lyapunov functiondefined as follows

V 12

e21 + e

22 + e

23 + e

2d + e

2e + e

2k + e

2l + e

2m + e

2c + e

2δ + e

2ρ1113872 1113873

(58)

which is a positive definite function on R11 e derivativewith respect to time of equation (58) gives us

_V e1 _e1 + e2 _e2 + e3 _e3 + ed _ed + ee _ee

+ ek _ek + el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

e1D1minus q1D

q1e1 + e2D1minus q2D

q2e2

+ e3D1minus q3D

q3e3 + ed _ed + ee _ee + ek _ek

+ el _el + em _em + ec _ec + eδ _eδ + eρ _eρ

(59)

Taking into account system (57) and equation (15)equation (59) becomes

_V e1D1minus q1 minus eee1 + ede3 minus h1D

q1minus 1e1 + v11113960 1113961

+ e2D1minus q2 minus ek y

22 minus y

211113872 1113873 minus el x

22 minus x

211113872 11138731113960

minus h2Dq2minus 1

e2 + v21113961 + e3D1minus q3 minus ece3 minus eδe11113960

minus eρe2 minus h3Dq3minus 1

e3 + v31113961 minus_ded minus _eee minus

_kek

minus_lel minus _mem minus _1113954cec minus

_δeδ minus _ρeρ

(60)

From equation (60) we deduce that the estimated pa-rameters update law is

12 Mathematical Problems in Engineering

_1113954d h4ed

_1113954e h5ee

_1113954k h6ek

_1113954l h7el

_1113954m h8em

_1113954c h9ec

_1113954δ h10eδ

_1113954ρ h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

where h4 h5 h6 h7 h8 h9 h10 and h11 are positive constantsFrom equation (60) we also deduce that the nonlinear

functions vi (i 1 2 3) are given by

v1 eee1 minus ede3

v2 ek y22 minus y

211113872 1113873 + el x

22 minus x

211113872 1113873

v3 ece3 + eδe1 + eρe2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(62)

By replacing equations (61) and (62) in (60) we get

_V minus h1e21 minus h2e

22 minus h3e

23 minus h4e

2d minus h5e

2e

minus h6e2k minus h7e

2l minus h8e

2m minus h9e

2c minus h10e

2δ

minus h11e2ρ lt 0

(63)

which is a negative definite function on R11 for positiveconstants hi (i 1 11) So we have found a functionwhich verifies the Lyapunov stability theorem (Vgt 0 _Vlt 0)

us the error functions converge globally and as-ymptotically towards zero for all initial conditions esynchronization of the states of the identical systems (51)and (52) is therefore complete So we have the followingresult

Theorem 4 5e identical financial systems (51) and (52)with unknown parameters are globally and asymptoticallysynchronized for all initial conditions by the adaptive controllaw (55) with vi (i 1 2 3) given by (62) and where theparameters update law is given by (61) and thehi (i 1 11) are positive constants

422 Simulation Results In this part we use the Adams-Bashforth-Moulton predictor-corrector method proposedby Diethelm et al [35] to solve the two identical fractionalorder systems (51) and (52) with the adaptive control law(55) the vi (i 1 2 3) given by (62) and the law forupdating the parameters is given by (61) For the simulationthe time-step h 0001 is chosen e initial conditions forthe master system are (x1(0) y1(0) z1(0)) (12 15 16)

and for the slave system (x2(0) y2(0) z2(0))

(02 05 06) are usede orders qi (i 1 2 3) are taken asfollows (q1 q2 q3) (1 088 1) ie the case of incom-mensurate order

e parameters of the new fractional order financialsystem are chosen as follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(64)

0 1 2 3 4 5 6 7 8 9 10

8

4

0

ndash4

x 1(t)

x2(t)

x1(t)

x2(t)

Time t

(a)

3

2

1

0

y 1(t)

y2(t)

y1(t)

y2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(b)

20

ndash2ndash4

z 1(t)

z2(t)

z1(t)

z2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(c)

2

1

0

ndash1e 1(t)

e2(t)

e 3(t)

e1(t)

e2(t)e3(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(d)

Figure 9 (andashc) Time evolutions of the master and slave systems state variables (x1 x2) (y1 y2) (z1 z2) respectively in fractional ordercase and (d) time evolution of the error functions e1(black line) e2(red line) and e3(blue line)

Mathematical Problems in Engineering 13

For hi (i 1 11) of the adaptive and update lawswe choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) 1 1113954k(0) minus 1 1113954c(0) 5 1113954m(0) 61113954l(0) 3 1113954ρ(0) 4 1113954d(0) 3 1113954δ(0) 7 (65)

By applying the adaptive control law (55) and the parameterupdate law (61) to the new controlled fractional-order financialsystem (52) the results of numerical simulations are shown inFigures 9(a)ndash9(d) and 10 From Figures 9(a)ndash9(c) it can be seenthat the state variables of the master and slave systems aresynchronized Figure 9(d) shows the error system whicheventually converges to zero Finally Figure 10 shows the es-timated parameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ which as it can beseen converge towards the real values of the parameters of thesystem ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(66)

5 Conclusions

In this paper the adaptive control and the adaptive syn-chronization of a new financial system with unknownconstant parameters were studied in the cases of integer andfractional order e adaptive control law and the adaptivesynchronization law were designed based on Lyapunovrsquosstability theory and on the adaptive control theory e lawshave been designed in the cases of integer and incom-mensurate fractional order system

e proposed adaptive control technique is effective forchaos control and synchronization of the new financial

system when the constant parameters of the system areunknown Numerical simulations are carried out to provethe efficiency of the control and synchronization techniquesdesigned in this work

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank IMSP-UAC and the German AcademicExchange Service (DAAD) for financial support under theprogramme ldquoIn-CountryIn-Region Scholarship Programmerdquo

References

[1] K B Oldham and J Spanier 5e Fractional Calculus Aca-demic Press New York NY USA 1974

[2] B Ross ldquoFractional calculus and its applicationsrdquo LectureNotes in Mathematics Vol 457 Springer-Verlag New YorkNY USA 1975

7

6

5

4

3

2

1

0

ndash1

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

ρhat = 005 = ρ δhat = 1 = δlhat = 1 = l ehat = 03 = e

Figure 10 Parameter estimates for adaptive synchronization in fractional order case

14 Mathematical Problems in Engineering

[3] V Daftardar-Gejji and S Bhalekar ldquoChaos in fractional or-dered Liu systemrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1117ndash1127 2010

[4] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons amp Fractals vol 36no 5 pp 1305ndash1314 2008

[5] K Rajagopal A Bayani S Jafari A Karthikeyan andI Hussain ldquoChaotic dynamics of a fractional order glucose-insulin regulatory systemrdquo Frontiers of Information Tech-nology amp Electronic Engineering vol 21 no 7 pp 1108ndash11182019

[6] T T Hartley C F Lorenzo and H Killory Qammer ldquoChaosin a fractional order Chuarsquos systemrdquo IEEE Transactions onCircuits and Systems I Fundamental 5eory and Applicationsvol 42 no 8 pp 485ndash490 1995

[7] Q Jia ldquoChaos control and synchronization of the Newton-Leipnik chaotic systemrdquo Chaos Solitons amp Fractals vol 35no 4 pp 814ndash824 2008

[8] M T Yassen ldquoAdaptive control and synchronization of amodified Chuarsquos circuit systemrdquo Applied Mathematics andComputation vol 135 no 1 pp 113ndash128 2003

[9] V Sundarapandian ldquoAdaptive control and synchronizationof a generalized Lotka-Volterra systemrdquo International Journalof Bioinformatics amp Biosciences vol 1 no 1 pp 1ndash12 2011

[10] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A StatisticalMechanics and Its Applications vol 389 no 12 pp 2434ndash2442 2010

[11] U E Kocamaz A Goksu H Taskin and Y UyarogluldquoControl of chaotic two-predator one-prey model with singlestate control signalsrdquo Journal of Intelligent Manufacturingpp 1ndash10 2020

[12] A Hajipour and H Tavakoli ldquoDynamic analysis and adaptivesliding mode controller for a chaotic fractional incommen-surate order financial systemrdquo International Journal of Bi-furcation and Chaos vol 27 no 13 p 14 2017

[13] X Yi R Guo and Y Qi ldquoStabilization of chaotic systems withboth uncertainty and disturbance by the UDE-based controlmethodrdquo IEEE Access vol 8 no 1 pp 62471ndash62477 2020

[14] L Liu B Li and R Guo ldquoConsensus control for networkedmanipulators with switched parameters and topologiesrdquo IEEEAccess vol 9 pp 9209ndash9217 2021

[15] T Hou Y Liu and F Deng ldquoFinite horizon H2∕Hinfin controlfor SDEs with infinite Markovian jumpsrdquo Nonlinear AnalysisHybrid Systems vol 34 pp 108ndash120 2019

[16] R Xu and F Zhang ldquo-Nash mean-field games for generallinear-quadratic systems with applicationsrdquo Automaticavol 114 pp 1ndash4 2020

[17] R HilferApplications of Fractional Calculus in Physics WorldScientific Hackensack NJ USA 2001

[18] R He and P G Vaidya ldquoImplementation of chaotic cryp-tography with chaotic synchronizationrdquo Physical Review Evol 57 no 2 pp 1532ndash1535 1998

[19] J H Park ldquoChaos synchronization of a chaotic system vianonlinear controlrdquo Chaos Solitons amp Fractals vol 25 no 3pp 579ndash584 2005

[20] L Huang R Feng andMWang ldquoSynchronization of chaoticsystems via nonlinear controlrdquo Physics Letters A vol 320no 4 pp 271ndash275 2004

[21] S Bhalekar and V Daftardar-Gejji ldquoSynchronization ofdifferent fractional order chaotic systems using active con-trolrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3536ndash3546 2010

[22] H N Agiza and M T Yassen ldquoSynchronization of Rosslerand Chen chaotic dynamical systems using active controlrdquoPhysics Letters A vol 278 no 4 pp 191ndash197 2001

[23] T-l Liao ldquoAdaptive synchronization of two Lorenz SystemsrdquoChaos Solitons amp Fractals vol 9 no 9 pp 1555ndash1561 1998

[24] H Yu G Cai and Y Li ldquoDynamic analysis and control of anew hyperchaotic finance systemrdquo Nonlinear Dynamicsvol 67 no 3 pp 2171ndash2182 2012

[25] J-h Ma and Y-s Chen ldquoStudy for the bifurcation topologicalstructure and the global complicated character of a kind ofnonlinear finance system Irdquo Applied Mathematics and Me-chanics vol 22 no 11 pp 1240ndash1251 2001

[26] M S Abd-Elouahab N E Hamri and J Wang ldquoChaoscontrol of a fractional order financial systemrdquo MathematicalProblems in Engineering vol 2010 Article ID 27064618 pages 2010

[27] I Hernandez C Mateos J Nuntildeez and A F Tenorio ldquoLietheory applications to problems in mathematical finance andeconomicsrdquo Applied Mathematics and Computation vol 208no 2 pp 446ndash452 2009

[28] S A David J A T Machado D D Quintino andJ M Balthazar ldquoPartial chaos suppression in a fractionalorder macroeconomic modelrdquo Mathematics and Computersin Simulation vol 122 pp 55ndash68 2016

[29] J Yang E Zhang and M Liu ldquoBifurcation analysis and chaoscontrol in a modified finance system with delayed feedbackrdquoInternational Journal of Bifurcation and Chaos vol 26 no 062016

[30] A Hajipour and H Tavakoli ldquoAnalysis and circuit simulationof a novel nonlinear fractional incommensurate order fi-nancial systemrdquo Optik vol 127 no 22 pp 10643ndash106522016

[31] X Zhao Z Li and S Li ldquoSynchronization of a chaotic financesystemrdquo Applied Mathematics and Computation vol 217no 13 pp 6031ndash6039 2011

[32] Y Liao Y Zhou F Xu and X-B Shu ldquoA study on thecomplexity of a new chaotic financial systemrdquo Complexityvol 2020 Article ID 8821156 5 pages 2020

[33] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

[34] I Petras Fractional-order Nonlinear Systems ModelingAnalysis and Simulation Springer Science amp Business MediaBerlin Germany 2011

[35] K Diethelm N J Ford andA D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquo Nonlinear Dynamics vol 29 no 1ndash4 pp 3ndash222002

Mathematical Problems in Engineering 15

_1113954d h4ed

_1113954e h5ee

_1113954k h6ek

_1113954l h7el

_1113954m h8em

_1113954c h9ec

_1113954δ h10eδ

_1113954ρ h11eρ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(61)

where h4 h5 h6 h7 h8 h9 h10 and h11 are positive constantsFrom equation (60) we also deduce that the nonlinear

functions vi (i 1 2 3) are given by

v1 eee1 minus ede3

v2 ek y22 minus y

211113872 1113873 + el x

22 minus x

211113872 1113873

v3 ece3 + eδe1 + eρe2

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(62)

By replacing equations (61) and (62) in (60) we get

_V minus h1e21 minus h2e

22 minus h3e

23 minus h4e

2d minus h5e

2e

minus h6e2k minus h7e

2l minus h8e

2m minus h9e

2c minus h10e

2δ

minus h11e2ρ lt 0

(63)

which is a negative definite function on R11 for positiveconstants hi (i 1 11) So we have found a functionwhich verifies the Lyapunov stability theorem (Vgt 0 _Vlt 0)

us the error functions converge globally and as-ymptotically towards zero for all initial conditions esynchronization of the states of the identical systems (51)and (52) is therefore complete So we have the followingresult

Theorem 4 5e identical financial systems (51) and (52)with unknown parameters are globally and asymptoticallysynchronized for all initial conditions by the adaptive controllaw (55) with vi (i 1 2 3) given by (62) and where theparameters update law is given by (61) and thehi (i 1 11) are positive constants

422 Simulation Results In this part we use the Adams-Bashforth-Moulton predictor-corrector method proposedby Diethelm et al [35] to solve the two identical fractionalorder systems (51) and (52) with the adaptive control law(55) the vi (i 1 2 3) given by (62) and the law forupdating the parameters is given by (61) For the simulationthe time-step h 0001 is chosen e initial conditions forthe master system are (x1(0) y1(0) z1(0)) (12 15 16)

and for the slave system (x2(0) y2(0) z2(0))

(02 05 06) are usede orders qi (i 1 2 3) are taken asfollows (q1 q2 q3) (1 088 1) ie the case of incom-mensurate order

e parameters of the new fractional order financialsystem are chosen as follows

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(64)

0 1 2 3 4 5 6 7 8 9 10

8

4

0

ndash4

x 1(t)

x2(t)

x1(t)

x2(t)

Time t

(a)

3

2

1

0

y 1(t)

y2(t)

y1(t)

y2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(b)

20

ndash2ndash4

z 1(t)

z2(t)

z1(t)

z2(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(c)

2

1

0

ndash1e 1(t)

e2(t)

e 3(t)

e1(t)

e2(t)e3(t)

0 1 2 3 4 5 6 7 8 9 10Time t

(d)

Figure 9 (andashc) Time evolutions of the master and slave systems state variables (x1 x2) (y1 y2) (z1 z2) respectively in fractional ordercase and (d) time evolution of the error functions e1(black line) e2(red line) and e3(blue line)

Mathematical Problems in Engineering 13

For hi (i 1 11) of the adaptive and update lawswe choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) 1 1113954k(0) minus 1 1113954c(0) 5 1113954m(0) 61113954l(0) 3 1113954ρ(0) 4 1113954d(0) 3 1113954δ(0) 7 (65)

By applying the adaptive control law (55) and the parameterupdate law (61) to the new controlled fractional-order financialsystem (52) the results of numerical simulations are shown inFigures 9(a)ndash9(d) and 10 From Figures 9(a)ndash9(c) it can be seenthat the state variables of the master and slave systems aresynchronized Figure 9(d) shows the error system whicheventually converges to zero Finally Figure 10 shows the es-timated parameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ which as it can beseen converge towards the real values of the parameters of thesystem ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(66)

5 Conclusions

In this paper the adaptive control and the adaptive syn-chronization of a new financial system with unknownconstant parameters were studied in the cases of integer andfractional order e adaptive control law and the adaptivesynchronization law were designed based on Lyapunovrsquosstability theory and on the adaptive control theory e lawshave been designed in the cases of integer and incom-mensurate fractional order system

e proposed adaptive control technique is effective forchaos control and synchronization of the new financial

system when the constant parameters of the system areunknown Numerical simulations are carried out to provethe efficiency of the control and synchronization techniquesdesigned in this work

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank IMSP-UAC and the German AcademicExchange Service (DAAD) for financial support under theprogramme ldquoIn-CountryIn-Region Scholarship Programmerdquo

References

[1] K B Oldham and J Spanier 5e Fractional Calculus Aca-demic Press New York NY USA 1974

[2] B Ross ldquoFractional calculus and its applicationsrdquo LectureNotes in Mathematics Vol 457 Springer-Verlag New YorkNY USA 1975

7

6

5

4

3

2

1

0

ndash1

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

ρhat = 005 = ρ δhat = 1 = δlhat = 1 = l ehat = 03 = e

Figure 10 Parameter estimates for adaptive synchronization in fractional order case

14 Mathematical Problems in Engineering

[3] V Daftardar-Gejji and S Bhalekar ldquoChaos in fractional or-dered Liu systemrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1117ndash1127 2010

[4] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons amp Fractals vol 36no 5 pp 1305ndash1314 2008

[5] K Rajagopal A Bayani S Jafari A Karthikeyan andI Hussain ldquoChaotic dynamics of a fractional order glucose-insulin regulatory systemrdquo Frontiers of Information Tech-nology amp Electronic Engineering vol 21 no 7 pp 1108ndash11182019

[6] T T Hartley C F Lorenzo and H Killory Qammer ldquoChaosin a fractional order Chuarsquos systemrdquo IEEE Transactions onCircuits and Systems I Fundamental 5eory and Applicationsvol 42 no 8 pp 485ndash490 1995

[7] Q Jia ldquoChaos control and synchronization of the Newton-Leipnik chaotic systemrdquo Chaos Solitons amp Fractals vol 35no 4 pp 814ndash824 2008

[8] M T Yassen ldquoAdaptive control and synchronization of amodified Chuarsquos circuit systemrdquo Applied Mathematics andComputation vol 135 no 1 pp 113ndash128 2003

[9] V Sundarapandian ldquoAdaptive control and synchronizationof a generalized Lotka-Volterra systemrdquo International Journalof Bioinformatics amp Biosciences vol 1 no 1 pp 1ndash12 2011

[10] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A StatisticalMechanics and Its Applications vol 389 no 12 pp 2434ndash2442 2010

[11] U E Kocamaz A Goksu H Taskin and Y UyarogluldquoControl of chaotic two-predator one-prey model with singlestate control signalsrdquo Journal of Intelligent Manufacturingpp 1ndash10 2020

[12] A Hajipour and H Tavakoli ldquoDynamic analysis and adaptivesliding mode controller for a chaotic fractional incommen-surate order financial systemrdquo International Journal of Bi-furcation and Chaos vol 27 no 13 p 14 2017

[13] X Yi R Guo and Y Qi ldquoStabilization of chaotic systems withboth uncertainty and disturbance by the UDE-based controlmethodrdquo IEEE Access vol 8 no 1 pp 62471ndash62477 2020

[14] L Liu B Li and R Guo ldquoConsensus control for networkedmanipulators with switched parameters and topologiesrdquo IEEEAccess vol 9 pp 9209ndash9217 2021

[15] T Hou Y Liu and F Deng ldquoFinite horizon H2∕Hinfin controlfor SDEs with infinite Markovian jumpsrdquo Nonlinear AnalysisHybrid Systems vol 34 pp 108ndash120 2019

[16] R Xu and F Zhang ldquo-Nash mean-field games for generallinear-quadratic systems with applicationsrdquo Automaticavol 114 pp 1ndash4 2020

[17] R HilferApplications of Fractional Calculus in Physics WorldScientific Hackensack NJ USA 2001

[18] R He and P G Vaidya ldquoImplementation of chaotic cryp-tography with chaotic synchronizationrdquo Physical Review Evol 57 no 2 pp 1532ndash1535 1998

[19] J H Park ldquoChaos synchronization of a chaotic system vianonlinear controlrdquo Chaos Solitons amp Fractals vol 25 no 3pp 579ndash584 2005

[20] L Huang R Feng andMWang ldquoSynchronization of chaoticsystems via nonlinear controlrdquo Physics Letters A vol 320no 4 pp 271ndash275 2004

[21] S Bhalekar and V Daftardar-Gejji ldquoSynchronization ofdifferent fractional order chaotic systems using active con-trolrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3536ndash3546 2010

[22] H N Agiza and M T Yassen ldquoSynchronization of Rosslerand Chen chaotic dynamical systems using active controlrdquoPhysics Letters A vol 278 no 4 pp 191ndash197 2001

[23] T-l Liao ldquoAdaptive synchronization of two Lorenz SystemsrdquoChaos Solitons amp Fractals vol 9 no 9 pp 1555ndash1561 1998

[24] H Yu G Cai and Y Li ldquoDynamic analysis and control of anew hyperchaotic finance systemrdquo Nonlinear Dynamicsvol 67 no 3 pp 2171ndash2182 2012

[25] J-h Ma and Y-s Chen ldquoStudy for the bifurcation topologicalstructure and the global complicated character of a kind ofnonlinear finance system Irdquo Applied Mathematics and Me-chanics vol 22 no 11 pp 1240ndash1251 2001

[26] M S Abd-Elouahab N E Hamri and J Wang ldquoChaoscontrol of a fractional order financial systemrdquo MathematicalProblems in Engineering vol 2010 Article ID 27064618 pages 2010

[27] I Hernandez C Mateos J Nuntildeez and A F Tenorio ldquoLietheory applications to problems in mathematical finance andeconomicsrdquo Applied Mathematics and Computation vol 208no 2 pp 446ndash452 2009

[28] S A David J A T Machado D D Quintino andJ M Balthazar ldquoPartial chaos suppression in a fractionalorder macroeconomic modelrdquo Mathematics and Computersin Simulation vol 122 pp 55ndash68 2016

[29] J Yang E Zhang and M Liu ldquoBifurcation analysis and chaoscontrol in a modified finance system with delayed feedbackrdquoInternational Journal of Bifurcation and Chaos vol 26 no 062016

[30] A Hajipour and H Tavakoli ldquoAnalysis and circuit simulationof a novel nonlinear fractional incommensurate order fi-nancial systemrdquo Optik vol 127 no 22 pp 10643ndash106522016

[31] X Zhao Z Li and S Li ldquoSynchronization of a chaotic financesystemrdquo Applied Mathematics and Computation vol 217no 13 pp 6031ndash6039 2011

[32] Y Liao Y Zhou F Xu and X-B Shu ldquoA study on thecomplexity of a new chaotic financial systemrdquo Complexityvol 2020 Article ID 8821156 5 pages 2020

[33] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

[34] I Petras Fractional-order Nonlinear Systems ModelingAnalysis and Simulation Springer Science amp Business MediaBerlin Germany 2011

[35] K Diethelm N J Ford andA D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquo Nonlinear Dynamics vol 29 no 1ndash4 pp 3ndash222002

Mathematical Problems in Engineering 15

For hi (i 1 11) of the adaptive and update lawswe choose hi 3

For the initial value of the estimated parameters weassume the following values

1113954e(0) 1 1113954k(0) minus 1 1113954c(0) 5 1113954m(0) 61113954l(0) 3 1113954ρ(0) 4 1113954d(0) 3 1113954δ(0) 7 (65)

By applying the adaptive control law (55) and the parameterupdate law (61) to the new controlled fractional-order financialsystem (52) the results of numerical simulations are shown inFigures 9(a)ndash9(d) and 10 From Figures 9(a)ndash9(c) it can be seenthat the state variables of the master and slave systems aresynchronized Figure 9(d) shows the error system whicheventually converges to zero Finally Figure 10 shows the es-timated parameters 1113954d 1113954e 1113954k1113954l 1113954m 1113954c 1113954δ and 1113954ρ which as it can beseen converge towards the real values of the parameters of thesystem ie

e 03 k 002 c 1 m 1 l 01 ρ 005 d 12 δ 1

(66)

5 Conclusions

In this paper the adaptive control and the adaptive syn-chronization of a new financial system with unknownconstant parameters were studied in the cases of integer andfractional order e adaptive control law and the adaptivesynchronization law were designed based on Lyapunovrsquosstability theory and on the adaptive control theory e lawshave been designed in the cases of integer and incom-mensurate fractional order system

e proposed adaptive control technique is effective forchaos control and synchronization of the new financial

system when the constant parameters of the system areunknown Numerical simulations are carried out to provethe efficiency of the control and synchronization techniquesdesigned in this work

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank IMSP-UAC and the German AcademicExchange Service (DAAD) for financial support under theprogramme ldquoIn-CountryIn-Region Scholarship Programmerdquo

References

[1] K B Oldham and J Spanier 5e Fractional Calculus Aca-demic Press New York NY USA 1974

[2] B Ross ldquoFractional calculus and its applicationsrdquo LectureNotes in Mathematics Vol 457 Springer-Verlag New YorkNY USA 1975

7

6

5

4

3

2

1

0

ndash1

Para

met

er es

timat

es

0 1 2 3 4 5 6 7 8 9 10Time

dhat = 12 = d

γhat = 1 = γmhat = 1 = m

khat = 002 = k

ρhat = 005 = ρ δhat = 1 = δlhat = 1 = l ehat = 03 = e

Figure 10 Parameter estimates for adaptive synchronization in fractional order case

14 Mathematical Problems in Engineering

[3] V Daftardar-Gejji and S Bhalekar ldquoChaos in fractional or-dered Liu systemrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1117ndash1127 2010

[4] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons amp Fractals vol 36no 5 pp 1305ndash1314 2008

[5] K Rajagopal A Bayani S Jafari A Karthikeyan andI Hussain ldquoChaotic dynamics of a fractional order glucose-insulin regulatory systemrdquo Frontiers of Information Tech-nology amp Electronic Engineering vol 21 no 7 pp 1108ndash11182019

[6] T T Hartley C F Lorenzo and H Killory Qammer ldquoChaosin a fractional order Chuarsquos systemrdquo IEEE Transactions onCircuits and Systems I Fundamental 5eory and Applicationsvol 42 no 8 pp 485ndash490 1995

[7] Q Jia ldquoChaos control and synchronization of the Newton-Leipnik chaotic systemrdquo Chaos Solitons amp Fractals vol 35no 4 pp 814ndash824 2008

[8] M T Yassen ldquoAdaptive control and synchronization of amodified Chuarsquos circuit systemrdquo Applied Mathematics andComputation vol 135 no 1 pp 113ndash128 2003

[9] V Sundarapandian ldquoAdaptive control and synchronizationof a generalized Lotka-Volterra systemrdquo International Journalof Bioinformatics amp Biosciences vol 1 no 1 pp 1ndash12 2011

[10] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A StatisticalMechanics and Its Applications vol 389 no 12 pp 2434ndash2442 2010

[11] U E Kocamaz A Goksu H Taskin and Y UyarogluldquoControl of chaotic two-predator one-prey model with singlestate control signalsrdquo Journal of Intelligent Manufacturingpp 1ndash10 2020

[12] A Hajipour and H Tavakoli ldquoDynamic analysis and adaptivesliding mode controller for a chaotic fractional incommen-surate order financial systemrdquo International Journal of Bi-furcation and Chaos vol 27 no 13 p 14 2017

[13] X Yi R Guo and Y Qi ldquoStabilization of chaotic systems withboth uncertainty and disturbance by the UDE-based controlmethodrdquo IEEE Access vol 8 no 1 pp 62471ndash62477 2020

[14] L Liu B Li and R Guo ldquoConsensus control for networkedmanipulators with switched parameters and topologiesrdquo IEEEAccess vol 9 pp 9209ndash9217 2021

[15] T Hou Y Liu and F Deng ldquoFinite horizon H2∕Hinfin controlfor SDEs with infinite Markovian jumpsrdquo Nonlinear AnalysisHybrid Systems vol 34 pp 108ndash120 2019

[16] R Xu and F Zhang ldquo-Nash mean-field games for generallinear-quadratic systems with applicationsrdquo Automaticavol 114 pp 1ndash4 2020

[17] R HilferApplications of Fractional Calculus in Physics WorldScientific Hackensack NJ USA 2001

[18] R He and P G Vaidya ldquoImplementation of chaotic cryp-tography with chaotic synchronizationrdquo Physical Review Evol 57 no 2 pp 1532ndash1535 1998

[19] J H Park ldquoChaos synchronization of a chaotic system vianonlinear controlrdquo Chaos Solitons amp Fractals vol 25 no 3pp 579ndash584 2005

[20] L Huang R Feng andMWang ldquoSynchronization of chaoticsystems via nonlinear controlrdquo Physics Letters A vol 320no 4 pp 271ndash275 2004

[21] S Bhalekar and V Daftardar-Gejji ldquoSynchronization ofdifferent fractional order chaotic systems using active con-trolrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3536ndash3546 2010

[22] H N Agiza and M T Yassen ldquoSynchronization of Rosslerand Chen chaotic dynamical systems using active controlrdquoPhysics Letters A vol 278 no 4 pp 191ndash197 2001

[23] T-l Liao ldquoAdaptive synchronization of two Lorenz SystemsrdquoChaos Solitons amp Fractals vol 9 no 9 pp 1555ndash1561 1998

[24] H Yu G Cai and Y Li ldquoDynamic analysis and control of anew hyperchaotic finance systemrdquo Nonlinear Dynamicsvol 67 no 3 pp 2171ndash2182 2012

[25] J-h Ma and Y-s Chen ldquoStudy for the bifurcation topologicalstructure and the global complicated character of a kind ofnonlinear finance system Irdquo Applied Mathematics and Me-chanics vol 22 no 11 pp 1240ndash1251 2001

[26] M S Abd-Elouahab N E Hamri and J Wang ldquoChaoscontrol of a fractional order financial systemrdquo MathematicalProblems in Engineering vol 2010 Article ID 27064618 pages 2010

[27] I Hernandez C Mateos J Nuntildeez and A F Tenorio ldquoLietheory applications to problems in mathematical finance andeconomicsrdquo Applied Mathematics and Computation vol 208no 2 pp 446ndash452 2009

[28] S A David J A T Machado D D Quintino andJ M Balthazar ldquoPartial chaos suppression in a fractionalorder macroeconomic modelrdquo Mathematics and Computersin Simulation vol 122 pp 55ndash68 2016

[29] J Yang E Zhang and M Liu ldquoBifurcation analysis and chaoscontrol in a modified finance system with delayed feedbackrdquoInternational Journal of Bifurcation and Chaos vol 26 no 062016

[30] A Hajipour and H Tavakoli ldquoAnalysis and circuit simulationof a novel nonlinear fractional incommensurate order fi-nancial systemrdquo Optik vol 127 no 22 pp 10643ndash106522016

[31] X Zhao Z Li and S Li ldquoSynchronization of a chaotic financesystemrdquo Applied Mathematics and Computation vol 217no 13 pp 6031ndash6039 2011

[32] Y Liao Y Zhou F Xu and X-B Shu ldquoA study on thecomplexity of a new chaotic financial systemrdquo Complexityvol 2020 Article ID 8821156 5 pages 2020

[33] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

[34] I Petras Fractional-order Nonlinear Systems ModelingAnalysis and Simulation Springer Science amp Business MediaBerlin Germany 2011

[35] K Diethelm N J Ford andA D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquo Nonlinear Dynamics vol 29 no 1ndash4 pp 3ndash222002

Mathematical Problems in Engineering 15

[3] V Daftardar-Gejji and S Bhalekar ldquoChaos in fractional or-dered Liu systemrdquo Computers amp Mathematics with Appli-cations vol 59 no 3 pp 1117ndash1127 2010

[4] W-C Chen ldquoNonlinear dynamics and chaos in a fractional-order financial systemrdquo Chaos Solitons amp Fractals vol 36no 5 pp 1305ndash1314 2008

[5] K Rajagopal A Bayani S Jafari A Karthikeyan andI Hussain ldquoChaotic dynamics of a fractional order glucose-insulin regulatory systemrdquo Frontiers of Information Tech-nology amp Electronic Engineering vol 21 no 7 pp 1108ndash11182019

[6] T T Hartley C F Lorenzo and H Killory Qammer ldquoChaosin a fractional order Chuarsquos systemrdquo IEEE Transactions onCircuits and Systems I Fundamental 5eory and Applicationsvol 42 no 8 pp 485ndash490 1995

[7] Q Jia ldquoChaos control and synchronization of the Newton-Leipnik chaotic systemrdquo Chaos Solitons amp Fractals vol 35no 4 pp 814ndash824 2008

[8] M T Yassen ldquoAdaptive control and synchronization of amodified Chuarsquos circuit systemrdquo Applied Mathematics andComputation vol 135 no 1 pp 113ndash128 2003

[9] V Sundarapandian ldquoAdaptive control and synchronizationof a generalized Lotka-Volterra systemrdquo International Journalof Bioinformatics amp Biosciences vol 1 no 1 pp 1ndash12 2011

[10] S Dadras and H R Momeni ldquoControl of a fractional-ordereconomical system via sliding moderdquo Physica A StatisticalMechanics and Its Applications vol 389 no 12 pp 2434ndash2442 2010

[11] U E Kocamaz A Goksu H Taskin and Y UyarogluldquoControl of chaotic two-predator one-prey model with singlestate control signalsrdquo Journal of Intelligent Manufacturingpp 1ndash10 2020

[12] A Hajipour and H Tavakoli ldquoDynamic analysis and adaptivesliding mode controller for a chaotic fractional incommen-surate order financial systemrdquo International Journal of Bi-furcation and Chaos vol 27 no 13 p 14 2017

[13] X Yi R Guo and Y Qi ldquoStabilization of chaotic systems withboth uncertainty and disturbance by the UDE-based controlmethodrdquo IEEE Access vol 8 no 1 pp 62471ndash62477 2020

[14] L Liu B Li and R Guo ldquoConsensus control for networkedmanipulators with switched parameters and topologiesrdquo IEEEAccess vol 9 pp 9209ndash9217 2021

[15] T Hou Y Liu and F Deng ldquoFinite horizon H2∕Hinfin controlfor SDEs with infinite Markovian jumpsrdquo Nonlinear AnalysisHybrid Systems vol 34 pp 108ndash120 2019

[16] R Xu and F Zhang ldquo-Nash mean-field games for generallinear-quadratic systems with applicationsrdquo Automaticavol 114 pp 1ndash4 2020

[17] R HilferApplications of Fractional Calculus in Physics WorldScientific Hackensack NJ USA 2001

[18] R He and P G Vaidya ldquoImplementation of chaotic cryp-tography with chaotic synchronizationrdquo Physical Review Evol 57 no 2 pp 1532ndash1535 1998

[19] J H Park ldquoChaos synchronization of a chaotic system vianonlinear controlrdquo Chaos Solitons amp Fractals vol 25 no 3pp 579ndash584 2005

[20] L Huang R Feng andMWang ldquoSynchronization of chaoticsystems via nonlinear controlrdquo Physics Letters A vol 320no 4 pp 271ndash275 2004

[21] S Bhalekar and V Daftardar-Gejji ldquoSynchronization ofdifferent fractional order chaotic systems using active con-trolrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 11 pp 3536ndash3546 2010

[22] H N Agiza and M T Yassen ldquoSynchronization of Rosslerand Chen chaotic dynamical systems using active controlrdquoPhysics Letters A vol 278 no 4 pp 191ndash197 2001

[23] T-l Liao ldquoAdaptive synchronization of two Lorenz SystemsrdquoChaos Solitons amp Fractals vol 9 no 9 pp 1555ndash1561 1998

[24] H Yu G Cai and Y Li ldquoDynamic analysis and control of anew hyperchaotic finance systemrdquo Nonlinear Dynamicsvol 67 no 3 pp 2171ndash2182 2012

[25] J-h Ma and Y-s Chen ldquoStudy for the bifurcation topologicalstructure and the global complicated character of a kind ofnonlinear finance system Irdquo Applied Mathematics and Me-chanics vol 22 no 11 pp 1240ndash1251 2001

[26] M S Abd-Elouahab N E Hamri and J Wang ldquoChaoscontrol of a fractional order financial systemrdquo MathematicalProblems in Engineering vol 2010 Article ID 27064618 pages 2010

[27] I Hernandez C Mateos J Nuntildeez and A F Tenorio ldquoLietheory applications to problems in mathematical finance andeconomicsrdquo Applied Mathematics and Computation vol 208no 2 pp 446ndash452 2009

[28] S A David J A T Machado D D Quintino andJ M Balthazar ldquoPartial chaos suppression in a fractionalorder macroeconomic modelrdquo Mathematics and Computersin Simulation vol 122 pp 55ndash68 2016

[29] J Yang E Zhang and M Liu ldquoBifurcation analysis and chaoscontrol in a modified finance system with delayed feedbackrdquoInternational Journal of Bifurcation and Chaos vol 26 no 062016

[30] A Hajipour and H Tavakoli ldquoAnalysis and circuit simulationof a novel nonlinear fractional incommensurate order fi-nancial systemrdquo Optik vol 127 no 22 pp 10643ndash106522016

[31] X Zhao Z Li and S Li ldquoSynchronization of a chaotic financesystemrdquo Applied Mathematics and Computation vol 217no 13 pp 6031ndash6039 2011

[32] Y Liao Y Zhou F Xu and X-B Shu ldquoA study on thecomplexity of a new chaotic financial systemrdquo Complexityvol 2020 Article ID 8821156 5 pages 2020

[33] I Podlubny Fractional Differential Equations AcademicPress New York NY USA 1999

[34] I Petras Fractional-order Nonlinear Systems ModelingAnalysis and Simulation Springer Science amp Business MediaBerlin Germany 2011

[35] K Diethelm N J Ford andA D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquo Nonlinear Dynamics vol 29 no 1ndash4 pp 3ndash222002

Mathematical Problems in Engineering 15