Physica A 197 (1993) 130-143 North-Holland

SDZ: 037%4371(93)EOO72-M

Periodicity in phase space

trajectories of chaotic systems in

Hiroshi Shibata and Ryuji Ishizaki De&wtment of Physics, Kyushu University 33, Fukuoka 812, Japan

Received 18 February 1993

We have extracted various time correlations in trajectories of chaotic systems in phase space. The time correlations in trajectories are calculated according to their local Lyapunov exponents. The time series is decomposed into the nonchaotic component and the chaotic component. The order-q power spectrum is used to extract such various time correlations. We have investigated the logistic map and the conservative standard map. Especially in the conservative standard map a trajectory is shown to have both periodic characteristics and chaotic characteristics.

1. Introduction

A statistical mechanical formalism of chaos has been developed in last few years [l-5]. Recently in the process of the development the new theoretical device “order-q power spectrum” was born [6-81. In this section the statistical mechanical formalism of chaos will be described schematically.

Let us take a time series {z+, i = 0, 1,2, . . .} . First the order-q moment M,(n) is defined as

(1)

which corresponds to the partition function and q is the inverse temperature. From M,(n) the free energy 4(q) is derived as

4(q) = /ii_ i log M,(n) . (2)

The internal energy u(q) and the specific heat x(q) are given by

u(q) = ; 4(q) (3)

0378-4371/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved

H. Shibata, R. Ishizaki I Periodicity in trajectories of chaotic system 131

and

(4)

respectively. By Laplace transformation of eq. (1) the entropy S(u) [9] is derived as

e -nS(u) - = dq M,(n) emUCqjqn = (S(u(q) - u,,)) , (9

where

u, E i

n-l

F0 ui * (6) I

(* * a) means a time average as introduced in eq. (1). If M,(n) is written as -e’(q) n there exists a Legendre transform between the free energy 4(q) and the entropy S(U) as

S(u) = 4 4) 4 - 44 4) 7 (7)

dS(u) 4’ du ’ (f-9

Then the new approach appears naturally to get correlations in time series. That is the order-q power spectrum Z,(w) defined as

Z,(o) = !@_ Ub; n) 6(u(q) - 4)) (Z(w; n> exp( 4 CYit ui>)

(S(u( q) - U”)) = fez M,(n) ’ ’ (9)

where

Z(w;n)= -& nfl u,eitw12. I=0

(10)

Z,(w) is a power spectrum made of the time series whose average is u(q). In this paper the original Z,(w) is improved. After all we want power spectra

of trajectories in phase space which are selected by their local Lyapunov exponents, because it is considered that the state of motion of the phase point reflects the local Lyapunov exponents [lo]. If the local Lyapunov exponent is small the motion of the phase point is nonchaotic locally. On the other hand, if

132 H. Shibata, R. Ishizaki I Periodic&y in trajectories of chaotic systems

the local Lyapunov exponent is large the motion of the phase point is chaotic locally. So the improved order -q power spectrum is defined as

Z,(o) = Iii_ (Ztw; n> @A(q) - A,)) = lim (Z(0; n) eq”An) Gmd - A,>) II--= w-4 (e 9 ’ (11)

where A,, is the local Lyapunov exponent which is defined as

(12)

Z(w; n) is defined with the successive locations of the trajectory in phase space. {Ai, i = 0, 1,2,. . .} are local expansion rates along the trajectory and A(q) is the u(q) such that the successive local expansion rates are used as time series. This modified order-q power spectrum expresses the time correlations in the trajectories in phase space which are selected by their local Lyapunov expo- nents. This paper is organized as follows. In section 2 the logistic map (X,,, = a - X:) is taken up. The control parameter a is chosen just at a critical point and just after. In section 3 the conservative standard map is taken up. The phases which we study are before the last KAM break, just after the break and a little far after the break. Conclusions are given in section 4.

2. Time correlations in the logistic map

Let us take up the logistic map

X 1+1 =a-Xf. (13)

This map has been studied extensively by many investigators. Just at a = a, = 1.401155189. . . ) the attractor becomes a critical 2” attractor and changes to a chaotic attractor [ 111.

First we study the logistic map at a = a,. A(q) is shown in fig. 1 where n is 512 and the number of ensembles is 240 000. Phase-q transition takes place at

q=q,= -0.8. Fig. 1 shows that A(0) =O, that is, the Lyapunov exponent equals zero. The absolute values of A(q) are small. And at a = a,, A(q) changes values from negative to positive continuously. The change is so small that the big change in Z,(o) does not happen as shown in fig. 2 where n is 512 and the number average over order-q power spectra is 2400. Of course {X,} are used as the locations of the trajectory when Z,(w)‘s are constructed. As the value of parameter q increases Z,(w) begins to exhibit a fine structure in the small peaks.

H. Shibata, R. Ishizaki I Periodicity in trajectories of chaotic systems 133

-4 0 4 Q

x 0.006

0.003

0

-4 0 4 Q

Fig. 1. A vs q (a) and ,y vs q (b) of the logistic map at a = a,. n is 512 and the number of ensembles is 240 000.

I~~~~~~ 0 1 2 3 0 1 2 3

q = -6.0 w (I = -3.0

1~~~~~~~~ I . . . . . . . . . . . . . . . . . . . ..-..-...... 1 .-_-_-_-_-_---_---.-_-_-_-_-~-_~

0 1 2 3 0 1 2 3 0 1 2 3

q = 0.0 q = 3.0 q = G.0

Fig. 2. Z,(o)% of the logistic map at a = a,. n is 512 and the number of ensembles is 2400.

134 H. Shibatu, R. Ishizaki I Periodic@ in trajectories of chaotic systems

Then a is set equal to a = a,(1 + 0.01). A(q) is shown in fig. 3. As a is in the chaotic regime A(q) is positive. The q-phase transition is a little more drastic than in the case a = a,. I,(w) changes from periodic to chaotic as shown in fig. 4. In the range q < 0, Z4(o)'s show a kind of periodicity and order. Just at

h

0.08

Fig. 3. A vs q (a) and ,y vs q (b) of the logistic

-4 0 4 4

X

0.06

0.03

0

-4 0 4

Q map at a = a,(1 + 0.01). n and the number

ensembles are the same as in fig. 1.

~

lo*

loo

10-a . .

tl. .* * . . . . . . I . . . . ,....#.A ,....t....a....t.

~~~~~~~~~

0 1 2 3 0 1 2 3 0 1 2 3

q = 0.0 9 = 3.0 q = 6.0

Fig. 4. Z,(o)% of the logistic map at a = a,(1 + 0.01). n and the number of ensembles are the same as in fig. 2.

of

H. Shibata, R. lshizaki / Periodicity in trajectories of chaotic systems 135

q = 0, Z,(w) takes a strange form like a Weierstrass function. In the range q > 0, Z4(o)‘s appear as their order is lost.

3. Time correlations in the conservative standard map

In this section let us take the conservative standard map [12,13]

’ (14)

where K is a nonlinearity parameter and the Jacobian is one. We study the conservative standard map at K= 0.5, K,(l + 0.13) and K,(l + 0.3), where Kc = 0.971635406. . . is the value at which the last KAM breaks up [14]. In this section II is 512 and the number average over the order-q power spectra is 3000, and the time series used in this section are made of a very long trajectory by cutting. {Z,} are used as the locations for Z,(w)%.

At K = 0.5, the phase-space portrait is shown in fig. 5. The initial position is located at (e,, J,) = (0.501,O.OOO). The static characteristics of local Lyapunov exponents appear in A(q) as shown in fig. 6. The slope of A(q) at the side where q is small is steep and the slope at the side where q is large is gentle. This makes us conjecture that where q is small, that is, A(q) is small, the trajectory behaves periodically and where q is large, that is, A(q) is large, the trajectory is chaotic, and the transition from the nonchaotic to the chaotic happens at the q, where x(q) has a peak. Fig. 7 shows the phase points of the trajectories whose local Lyapunov exponents A, take values between 0.005 and

0.5 e 1

Fig. 5. Phase-space portrait of the conservative standard map at K = 0.5. The initial position of (0, J) is (0.501,O.OOO) and the twenty thousand points are depicted.

136 H. Shibata, R. Ishizaki I Periodic@ in trajectories of chaotic systems

A

-0.2 0 0.2 !l

x

-0.2 0 0.2

Q

Fig. 6. A vs q (a) and x vs q (b) of the conservative standard map at K = 0.5. n and the number of

ensembles are the same as in fig. 1.

Fig. 7. Phase points which satisfy the condition 0.005 < A, < 0.010 at K = 0.5. n is 512.

0.010. These trajectories are produced from the very long trajectory whose initial point in the phase space is (0.501,O.OOO) by cutting. The phase points stick to the phase boundary. Fig. 8 confirms the conjecture. Where q < q, =

0.04, the Z,(w)% show some peaks. Peaks are at w = HIT and An. The periodic modes are picked up clearly and an o = 0 mode is not found. Where q > q, the back ground raises. This means that the order-q power spectra show chaos. Another evidence that the trajectory becomes chaotic is present. The peak at w = 0 grows higher. This means that the mode whose period is infinite comes out. The heights of special peaks at w # 0 become lower and the half widths of

H. Shibata, R. Ishizaki I Periodicity in trajectories of chaotic systems 137

q = 0.0

q = -0.12

q = 0.04

Q = -0.04

q = 0.12

q = 0.2

Fig. 8. I,(o)‘s of the conservative standard map at K = 0.5. n is 512 and the number of ensembles is 3000.

them become broader. The phase point in phase space is seen to move only in the chaotic sea at random. However, the fact that the phase point moves sometimes periodically to stick to the phase boundary is remarkable.

At K = K,(l + 0.13), the phase point fills up the phase space slowly as shown in fig. 9. The static characteristics are shown in fig. 10. Also at this parameter A(q) has the same tendency as in the case K = 0.5. Fig. 11 shows the phase points of the trajectories whose local Lyapunov exponents are between 0.01 and 0.03 made in the same way as fig. 7. The phase points stick

138 H. Shibata, R. Ishizaki I Periodic@ in trajectories of chaotic systems

0.5 e 1

Fig. 9. Phase-space portrait of the conservative standard map at K = K,(l + 0.13). The initial position of (13, J) and the number of points are the same as in fig. 5.

-0.2 0 0.2 Q

-0.2 0 0.2

Q

Fig. 10. A vs q (a) and ,y vs q (b) of the conservative standard map at K = K,(l + 0.13). n and the number of ensembles are the same as in fig. 1.

around the big torus and the other tori. At the time the trajectory behaves periodically as shown in fig. 12. Where q < q, = 0.0, the ZJw)‘s have a sharp peak at w = $IT. The background is low and has no peaks at w = 0. Where q > q,, ZJ.(o)‘s have a large peak at w = 0. And the background around o = 0 is high. In this range where l = (K - KJIK, > 0 the periodicity and chaos are found clearly irrespective of the diffusion.

At K = Z&(1 + 0.3), the phase point fills up the phase space relatively fast as shown in fig. 13. The static characteristics are shown in fig. 14. The tendency of

139

Fig. 11. Phase points which satisfy the condition 0.01 <A, CO.03 at K = K,(l + 0.13). n is 512.

.: . . ” f 2.. ” .*. .“‘*‘.

. 0 1 1 :. a*.-

0 1 2 3 q = -0.2 w

__I *N c

q = 0.0

. . . . :.: ~I

J . .-. 8.. b.. . I..

q = -0.12

1 ‘> 8.. 88

,iA 1 . I . ‘

q = 0.04

., - ‘*

q = -0.04

1

I

;.. , ;, .. , . ,+,j

q = 0.12

q = 0.2

Fig. 12. Z,(o)% of the conservative standard map at K = K,(1+0.13). n and the number of ensembles are the same as in fig. 7.

140 H. Shibata, R. Ishizaki I Periodic@ in trajectories of chaotic sysrems

Fig. 13. Phase-shift portrait of the conservative standard map at K = K,(l f 0.3). The initial position of (0, J) and the number of points are the same as in fig. 5.

A X

O’* : I.“‘!““,’

(a)

0 C’ “‘1. ” ” ” ” 1 'I " 1 ,- -0.2 0 0.2 -0.2 0 0.2

Q Q Fig. 14. A vs q (a) and ,y vs q (b) of the conservative standard map at K = K,(l + 0.3). n and the number of ensembles are the same as in fig. 1.

the slope is similar to the two previous cases. The phase points of the trajectories whose local Lyapunov exponents are between 0.01 and 0.03, are shown in fig. 15. Figs. 15 and 16 show that the trajectory behaves periodically when it sticks around the big torus. Z,(o)% are shown in fig. 16. Where

4<4, = -0.02, Z,(w)% have a peak at o = f m. It is never seen in the ordinary power spectrum. Where q > q,, the background raises and the peak at w = 0 is high. The special peak at o = $a becomes a gentle hill.

0 0.5 e ’

141

Fig. 15. Phase points which satisfy the condition 0.01 < A, < 0.03 at K = K,(l + 0.3). n is 512.

ZP 2

o_. E :.

1

0 1 2 3 q = -0.2 w

q = 0.0

Fig. 16. Z,(o)‘s of the conservative ensembles are the same as in fig. 7.

. . ,....I....I....I.

q = -0.12

1 ‘L q = 0.04

-.

q = -0.04

q = 0.12

standard map at K = K,(l + 0.3). n and the number of

142 H. Shibata, R. Ishizaki I Periodic@ in trajectories of chaotic system

4. Conclusions

As shown in sections 2 and 3 the order-q power spectrum is very useful in the case that various modes are mixed. The key point is that time series of the phase point are selected by the local Lyapunov exponents. The reason why the time series are selected this way is that there is the tendency that the trajectories are periodic locally when local Lyapunov exponents are small and they are chaotic locally when local Lyapunov exponents are large. But in section 2 the order-q power spectra are not understood fully. At the critical point in the logistic map the mean of the fine structures in the Z,(w)% cannot be understood. It might be thought that the larger the local Lyapunov exponents the complexer the motion of a phase point. So Z,(o)‘s become complex where q > q,. But as the original trajectory is 2” periodic, no matter how complex the structure in Z,(w) becomes, the order survives.

It is believed that in Hamiltonian systems the sticking motion to tori of a phase point continues for a very long time [15-171 and the irregular motion in the chaotic sea continues for a very long time. In the present study it is made clear that the simple period motion sticking to tori and the irregular motion in the chaotic sea interchange within a relatively short time. The average time staying in one of the two states is measured by the maximum value of the slope of A(q). The values in the present case are about 1.6, 2.8 and 4.3. This short time scale transition from nonchaotic to chaotic is our finding.

Recently one of the authors has developed a multi-order power spectrum [lS]. This can be used in many stochastic processes. Many other findings may be found through the new type of power spectrum.

Acknowledgements

We would like to thank all members of our group at Kyushu University for stimulating discussions.

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