Chaotic Circuit Using a Colpitts Oscillator and a Resonator

Hayato Amo1, Yasuteru Hosokawa1 and Yoshifumi Nishio2

1 Shikoku UniversityFurukawa, Ohjin, Tokushima, Japan

Phone:+81-88-665-1300FAX:+81-88-637-1318

E-mail: s002337004@edu.shikoku-u.ac.jpE-mail: hosokawa@keiei.shikoku-u.ac.jp

2 Tokushima University2-1 Minami-Josanjima, Tokushima, Japan

Phone:+81-88-656-7470FAX:+81-88-656-7471

E-mail: nishio@ee.tokushima-u.ac.jp

Abstract

In this study, a novel chaotic circuit is proposed. The circuitconsists of a Colpitts oscillator, a resonator and a diode. Byusing a idealized diode model, exact solutions are derived.Chaotic attractors are shown in circuit experiments and com-puter calculations of exact solutions.

1. Introduction

Chaos can be observed various field of the natural world.Therefore, many researchers pay attention to this phe-nomenon and related phenomena. In a field of electric en-gineering, chaotic phenomena have been investigated using achaotic circuit. Elements of an electric circuit can be obtainedin very low cost and the specific is very high. Additionally,a chaotic circuit has a simple structure. Thus, it is easy togenerate chaotic phenomena on an electric circuit.

However, designing a novel chaotic circuit is difficult ex-cept experts like [1]–[3] . A designing method is proposedin [4]. This method is that coupling an oscillator and a res-onator with diodes as shown in Fig. 1. The advantage is thatthe method is simple. Therefore, this method is very use-ful method for many researchers. On the other hand, thisadvantage is that there is no rule for setting the parameters.Breaking the disadvantage is essential to make this methodperfected.

In this study, a novel chaotic circuit applied this methodis proposed. The circuit consists of a Colpitts oscillator, aresonator and a diode. Developing many sample circuits is animportant task for making this method perfected.

2. Circuit Model

Figure 2 is a proposed circuit in this study. A Colpitts oscil-lator connects to a resonator via a diode. In this model, all el-ements except a diode are modeled as linear elements. Diodemodel is shown in Fig. 3. Its v-i characteristics is shown inFig. 4. This model is an idealized model. Using this model,

ResonatorOscillator

Figure 1: Designing method for a chaotic circuit.

vi

1 C1

1 L1 v2 C2 C3

L2

v3

i2

R 1

R 2

R 3

f(v - v )2 1

Figure 2: Proposed system.

the analysis becomes easily. In order to investigate validity ofthis model, a circuit equation is derived.

(C1 + C2)dv1dt

= − 1

R1(v1 + Vth)− i1 − i2,

C3dv3dt

= − 1

R3

v3 +

R2

R1(v1 + Vth)

+ i2,

L1di1dt

= v1,

L2di2dt

= v1 − v3 + Vth,

(1)

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vi

Vth

ON

OFF

Figure 3: Idealized diode model.

Vthv

i

ON

OFF

Figure 4: v-i characteristic of the idealized diode model asshown in Fig. 3.

C1

dv1dt

= −i1,

L1di1dt

= v1,(2)

C2dv2dt

=C1

C2

(− 1

R1v2 − i2

),

C3dv3dt

= − 1

R3

(v3 +

R2

R1v2

)+ i2,

L2di2dt

= v2 − v3.

(3)

By substituting the normalized variables and the parameters,

xn =vnVth

for n = 1, 2, 3,

xn =

√L1

C1

in−3

Vthfor n = 4, 5,

d

dt= ” · ”, τ =

1√L1C1

t, α =1

R1

√L1

C1,

β =R2

R3, γ =

1

R2

√L1

C1, δ =

L1

L2and

ε =C1

C2=

C1

C3.

(4)

Equations (1), (2) and (3) are normalized asx1 =

ε

ε+ 1(−αx1 − x4 − x5 − α),

x3 = ε(−αβx1 − βx3 + x5 − αβ),

x4 = x1,

x5 = δ(x1 − x3 + 1),

(5)

x1 = x4,

x4 = x1,(6)

x2 = ε(−αx2 − x5),

x3 = ε(−γx3 − αβx2 + x5),

x5 = δ(x2 − x3),

(7)

We define two piecewise-linear regions as follows.

R1 ≡ (x1, x3, x4, x5)|x1 + i1 > 0

R2 ≡ (x1, x2, x3, x4, x5)|x2 − x1 − 1 < 0(8)

These regions are corresponding to the two state of the diode.Note that the circuit equations in R2 are completely decou-pled into two and three dimensional equations. In order toderive exact solutions, eigenvalues in two regions are derived.The characteristic equation of the circuit equations in eachlinear region is given as follows.In R1:

(ε+ 1)m4 + ε(α+ β + βε)m3 + ε(αβε+ 2δ+δε+ 1)m2 + ε2(αβδ + αδ + βδ + β)m+ δε2

= 0.(9)

In R2: For the resonator,

m2 + 1 = 0. (10)

For the Colpitts Oscillator,

m3 + ε(α+ γ)m2 + ε(αγε+ 2δ)m+ δε2(α+ αβ+γ) = 0.

(11)

The eigenvalues in each region can be calculated from thecharacteristic equations ( 9 ) - ( 11)

From circuit experiment and computer simulation usingEq. (5) - (7), The eigenvalues are calculated as follows:

R1 : λ1, λ2, σ ± jω

R2(Resonator) : ± j

R2(Colpitts Osc.) : λ, σ1 ± jω1

(12)

Further, the equilibrium points in R1 is described as fol-lows:

B = [ − α− β − αβ 0 0 0 ]T . (13)

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These values are calculated by making the right-hand sideof Eq. (5) to be equal to zero. The equilibrium point in R2 isthe origin. Then, we can describe the exact solutions in eachlinear region as follows.In R1:

x(τ)−B = F(τ) · F−1(0) · (x(0)−B). (14)

In R2:For the resonator,

xa(τ) = Ga(τ) ·Ga−1(0) · (xa(0)). (15)

For the Colpitts osc.,

xb(τ) = Gb(τ) ·Gb−1(0) · (xb(0)). (16)

In order to derive the Poincare map, Let us define the follow-ing subspace

S = S1 ∩ S2. (17)

where

S1 : x1 = −x4 =ε(−αx1 − x4 − x5 − α)

(ε+ 1).

S2 : x2 − x1 = 1.(18)

The subspace S1 corresponds to the boundary condition be-tween R1 and R2, while the subspace S2 corresponds to theON state of the diode. Namely, S corresponds to the transi-tional condition from R1 to R2. Let us consider the solutionstarting from an initial point on S. The solution returns backto S again after two subspaces as shown in Fig. 5. Hence, thePoincare map can be derived as follows:

T : S → S, x 7→ T(x0). (19)

where x0 is an initial pont on S, while T(x0) is the pintat which the solution starting from x0 hits S again. ThePoincare map can be represented as a composit map of thesubmaps T1 and T2. Namely, the Poincare map can be ob-tained as

T = T1 T2. (20)

3. Circuit Experiments and Computer Calculations

Figure 6 shows the circuit experimental results and com-puter calculated results. In the circuit experiments, parame-ters are set as R1 = 1 [kΩ], R3 = 200 [Ω], C1 = 0.10 [µF],C2 = C3 = 0.022 [µF], L1 = 200[mH] and L1 = 30[mH].In the computer calculations, parameters are set as α = 1.41,γ = 4.71, δ = 6.67 and ε = 4.5. R2 and β are set as con-trol parameters. A periodic orbit is observed in Fig. 6 (1)and (5). Chaotic attractors are observed in Fig. 6 (2), (3) and

R 2R 1

T1 T2

S

Figure 5: Route map.

(4). These results shows this model does not lose importantfeatures of the original circuit related with the generation ofchaos. Additionally, the nonlinear element is only the diode.It is mean that the Colpitts oscillator plays a role of expand-ing and the diode play a role of folding. These are known asthe essence of generating chaos.

4. Conclusion

In this study, the novel chaotic circuit has been proposed.By applying a idealized diode model, exact solutions of eachregions are derived. Additionally, the Poincare map is de-fined. In circuit experiments and computer calculations,chaotic phenomena are observed.

In our future works, calculating the largest Lyapunov ex-ponents, the circuit analysis using exact solutions and so onwill be carried out.

References

[1] L. O. Chua, M. Komuro and T. Matsumoto, “The doublescroll family,” IEEE Trans. Circuits Syst., vol. 33, no. 11,pp. 1073–1118, 1986.

[2] N. Inaba, S. Mori and T. Saito, “Chaotic Phenomena in3rd Order Autonomous Circuits with a Linear NegativeResistance and an Ideal Diode II,” Technical Report ofIEICE, Vol. 87, No. 271, pp.9-15, Nov. 1987.

[3] M. Shinriki, M. Yamamoto and S. Mori, “Multimode os-cillations in a modified van der Pol Oscillator Containinga Positive Nonlinear Conductance,” Proc. IEEE, vol. 69,pp. 394–395, 1981.

[4] Y. Hosokawa, Y. Nishio and A. Ushida, “A DesignMethod of Chaotic Circuits Using an Oscillator and aResonator,” Proceedings of IEEE International Sympo-sium on Circuits and Systems (ISCAS’01), vol. 3, pp. 373-376, 2001.

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(1)

R2 = 8.60[kΩ] β = 24.5

(2)

R2 = 8.80[kΩ] β = 29.5

(3)

R2 = 10.1[kΩ] β = 33.0

(4)

R2 = 11.3[kΩ] β = 39.5

(5)

R2 = 12.0[kΩ] β = 41.0(a) (b) (c)

Figure 6: Circuit Experimental Results and Computer Calculation Results. (a) Circuit Experiments. Horizontal axis: v1 ( 1.0[V/div.] ). Vertical axis: v2 ( 2.0 [V/div.] ). (b) Attractors. Horizontal axis: x1. Vertical axis: x2. Gridlines show 2.5. (c)Poincare maps of (b). Horizontal axis: x1. Vertical axis: x2.

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