Local exponential divergence plot and optimal embedding of a chaotic time series

Transcription

1 PhysicsLettersA 181 (1993) North-Holland PHYSICS LETTERS A Local exponential divergence plot and optimal embedding of a chaotic time series Jianbo Gao and Zhemin Zheng Laboratoryfor Nonlinear Mechanics ofcontinuous Media, Institute ofmechanics, Chinese Academy ofsciences, Beijing , China Received 2 October 1992; revised manuscript received 26 July 1993; accepted for publication 4 August 1993 Communicated by A.R. Bishop We propose here a local exponentialdivergence plot which is capable ofproviding a new means ofcharacterizing chaotic time series. The suggested plot defines a time dependent exponent A and a plus exponenta which servesas a criterion for estimating simultaneously the minimal acceptable embedding dimension, the proper delay time and the largest Lyapunov exponent, In recent years much progress has been made in pling time & and construct vectors of the form understanding and characterizing chaotic dynamical X~=(x 1, X +L,..., X~(rn I )L), with m the embedding systems. Much ofthe progress has been brought about dimension and L delay time. Hence a dynamics F: by the important discovery that an appropriate state X1 *X~+1is defined, which is assumed to be represpace can be reconstructed from a scalar time series sentative of the original system. The distance be- [1]. Calculation of the correlation dimension and of tween X~and X~,denoted by dis(x~,x), is mapped the K2 entropy by the Grassberger Procaccia algo- to dis(x,+k, ~ k) after k iterations of F. The local rithm [2], and estimation of the Lyapunov expo- exponential divergence plot is defined by plotting nents [3 6] have become standard procedures for ln[dis(x,+k, A+k)/dis(X, X~)]versus ln[dis(x1, analyzing chaotic signals. However, one may not gain Xi)] when dis(x,, X~)is smaller than a prescribed much understanding by routine calculations under small distance r*. If we assume that most of these certain circumstances, since there are difficulties in sufficiently small distances dis(x~,x~)can be reinterpreting correlation integral results [71,which garded as distances between orbits, then if the moare intimately related to the problem of how to dis- tion is truly chaotic, points with dis(xa+k, XJ+k)> tinguish chaos from stochastic processes. Therefore, dis(x1, X~)will dominate and lie above the zero level it would be very helpful if a geometric method could line in the plot. be devised to view the dynamics, especially the local Figure 1 shows divergence plots with different m exponential divergence dominated behaviorof a time and L for the Rössler attractor (a = 0.15, b = 0.20, series, so that a glance at this divergence plot would c= 10.0, &t= 7t/25, the dynamics is reconstructed provide some insight into the dynamic system. from the x component of the flow). We will show We report here a kind of local exponential diver- below that the difference between these plots gives gence plot which enables one to view the dynamics a hint to optimal embedding, and m =3, L =8 coron a chaotic attractor. The simple plot provides a respond to optimal parameter values. The zero level criterion for the selection of the minimal acceptable line is added to fig. lb for a clear view of the diverembedding dimension and an optimal delay time. gence dominated behavior. When the unstable motion on the chaotic attractor A problem of significant practical importance is to only is extracted, a proper estimation of the largest determine the minimum acceptable embedding dipositive Lyapunov exponent can also be obtained. mension m~.a basic idea is that in the passage from Assume we have a time series x1, x2,..., with sam- dimension m to m + 1 one can differentiate between /93/S Elsevier Science Publishers B.V. All righjs reserved. 153

2 Volume 181, number 2 PHYSICS LETTERS A 4 October (a) m=2,l=9 6 (b) m3l=8 5 ~- :3 - o ~- ~ (c) rn~,l= ~ ~J6-~ 24 (6) rn=3,l=16 :3 -~ ~ ~ -~6 4 1n(dis(X~,X~)) Fig. 1. Local exponential divergence plots for the Rdssler attractor, k=9. points on the orbit X( n) that are true neighbors The key to selecting a proper delay time L is that the and points which are false neighbors pointswhich orbital motion should be as uniform as possible, and appear to be neighbors because the orbit is being distortion be small. This can be achieved by requirviewed in too small an embedding space. Based on ing that the number of points In[dis(X,-+k, ~+k)/ this basic idea, several methods are now available dis (X1, Xi)] with excessively large positive values in [8 11], which differ in implementations either by the divergence plot be as small as possible and the way of graphic display or by defining some appro- structure of the divergence plot be as compact as priate statistical quantity. When the embedding di- possible. This is the reason that the structure of fig. mension is increased from me ito m~,the structure lb is preferred to fig. lc or id. Actually, fig. lb is ofthe graphic representation and/or the value ofthe representative of that constructed from the original statistical quantity will undergo a radical change system and does not change much when m is further while further increasing m causes little change. Our increased. divergence plot implements this basic idea dynam- For a quantitative description, we define the time ically. When the embedding space is too small, the dependent divergence exponent A by ill-defined dynamics F and the false neighbors will generate many points of ln[dis(x~+k, XJ+k)/dis(Xi, A(k, m, L) = <ln[dis(x~+k, Xj+k)/diS(Xi, Xi)]> Xi)] with excessively large positive values in the di- (1) vergence plot. This is clearly shown by the difference between figs. la and lb. with dis(x,, X~,)-~r 8. The evolution time corre- Figure 1 also points out how to select the proper sponding to k is k& and the angular brackets denote delay time L. Dynamically, whenl is eithertoo small the ensemble average of all possible pairs (X 1, Xi-). or too large the dynamics F will not be very well de- Points with positive values of ln[dis(x1+k, XJ-,~k)/ fined, in the sense that excessively large values of dis(x,., X3)] are especially important since we are ln[dis(x~+k, A~,+k)/dis(XI,Xi)] frequently appear. more concerned with points ln[dis(x1 k, Xj+k)/ 154

3 Volume 181, number 2 PHYSICS LET FERS A 4 October m2 m=2 m=3~ I m=3 m=4~ m=4 0.0 I I C C C Delay Time L Delay Time L Fig. 2. A(L) and A+ (L) curves with different m for the Rössler attractor, k= 9. state space can be stated as follows. It is required that 1.6 * m=3,l=8 (solid line) Fbe a continuous mapping preserving neighborhood o m=3,l=6 relations. The minimal acceptable embedding dimension me is determined by requiring that the m=4,l=6 * m=8,l=4 structure of the divergence plot no longer changes radically, or equivalently, that A and A+ do not de- ~ crease significantly by further increasing m. When m is thus selected, for a series of L, the minima of L I / A + (L) and A (L) determine an optimal delay time. The physical significance of the quantity A is obvious. When the evolution time k& is very small, A/k& is the mean value of the so-called largest local Lyapunov exponent [12,13]. After several itera- tions, the separation vector between X 0.01* I I I 1 and X~will align with the eigendirection for the largest positive Evolution time k Lyapunov exponent, and A/k& is equivalent to the Fig. 3. A(k) curves with different m and L for the Rbssler standard estimation of this exponent. Hence when attractor, the proper reconstruction of the state space has been achieved, we would require that the A (k) curve for dis(x~,x~)i with excessively large positive values. k& not very small be a straight line which passes Thus we also define the plus exponent A + by through the origin when extrapolated. Otherwise, different values ofthe largest Lyapunov exponentwill A~(k, m, L) be obtained with different selections of k. This kind = <ln [dis (X,+k, A5+k) /dis(x1, X~)]+>, (2) of situation has been observed by Zeng et al. [6], however. Therefore, in practice at least we would rewhere + simply denotes that points with positive quire that A(k) depend linearly on k for k~(kmjn, values of ln[dis(x~+k, A+k)Idis(Xa, X,)] only are k,,~), and the interval (k,~~~kmjn)öt be not too averaged. In the following discussion, when only one small. The largest positive Lyapunov exponent can variable is considered, we will simply write 4(k), be objectively estimated by [A(k1 ) A (k~)]/ A(L) anda+(l) for convenience. (k1 k2)&, with k1, k2e(kmi~,kmax). Now the problem of properly reconstructing the Let us continue to discuss the Rössler system. Fig- 155

4 Volume 181, number 2 PHYSICS LETTERS A 4 October 1993 Table I Simultaneous minimal embedding dimension m and optical delay time L, and the largest Lyapunov exponent A. The total number of points N is given in the table for different model systems. System Optimal embedding parameters A others present results (N=2000) others present results Henon map 115] (a=l.4,b=0.3) m=2 m=2,l=l [5] 0.421±0.003 (N=5000) [9] (N=2000) Rössler [16] (bt=7t/25,a=0.l5, m=3,l=7 m=3,l= [5] 0.067±0.006 b=0.20,c= 10.0) (N=l0000) [10] (N=2000) Lorenz [17] (bt=0.03, a=lo, m=3, L= ] 1.48±0.03 b=~,r=45.92) (N=3000) Mackey Glass [18] (6t= 1.5, a=0.2, m=4, L&=0.3F m=4,lbt= [4] ± b=0.l,c=l0,f=30) (N=9000) [10] (N=4000) :3 (a) w=i (b) w= n(dis(x,,x~)) Fig. 4. Original (w 1) and modified (w= 54) local exponentialdivergence plots for the Lorenz system constructed from the x component ofthe flow, m=3,l=27, k=30. ure 2 shows the A (L) and A (L) curves, and we see punov exponent from the slope of the linear A (k) that the combination of m=3 and L=8 is an opti- curve of m=3 and L=8 is These results, tomal choice. Note that improper embedding (under- gether with results for other known model chaotic estimated m or improper L) always results in over- systems, are summarized in table 1. estimated positive exponent. Figure 3 gives the A (k) We note that, with regard to the proper reconstruccurves for different m and L. We see that when m =3, tion of the state space, the results of refs. [9,10] can the 4(k) curve for L = 8 shows a clear linear depen- be easily obtained by our method with a very small dence, while curves for the smaller value of L =6 or data set. Our method has the additional advantage the larger value of L = 12 (approximately the opti- that the approach is simpler, more natural and easier mal value suggested by ref. [141) are less satisfac- to understand and implement, and capable of protory. Figure 3 also shows two curves for m = 4, L=6 viding more information. and m = 8, L = 4, with improved linearity. Note that Finally we examine the assumption that most sufthe degree of the linearity is saturated when m is ficiently small distances dis(x 1, X1) can be regarded increased to 4, while the minimum acceptable value as distances between orbits. A problem related to this of m is 3. The estimated value of the largest Lya- is that whether and how the divergence plot for a 156

5 Volume 181, number 2 PHYSICS LETI ERSA 4 October 1993 Expression (1) corresponds to w= 1. Theiler [20] 4.0 has proposed a similar improvement to the calculation of the correlation dimension, and suggests w to be selected as the auto-correlation time. In our (b)w=54 -/,~~1I 1(a)w putting w equal to the embedding window (m 1 )L is already selected. does However, the job. we have tested numerically that case, Figure theoretically, 4b is a modified it does divergence not matter if plot. a very It can bigbe w seen easily that the curve-like part of fig. 4a, espe cially the part corresponding to very small distances Sb is the modified A(k) curve. It is now nearly un I I where ear. givesactually the statistics correct theislyapunov slope poor, of largely the exponent. modified suppressed. Thus A (k) wefigure curve know that the heavy black part in the divergence plot orig- Evolution Time k mates from the unstable motion, and a linear 4(k) Fig. 5. A(k) curves corresponding to fig. ~t. curve is a property of the unstable motion and characteristic of chaotic motions. continuous system changes with the sampling time Let us summarize. Two kinds of motion, tangen- &. Let us discuss the Lorenz system with a much tial motion and unstable motion, can be discerned smaller & of than the one taken in table 1. Fig- from the structure ofthe divergence plot. The former ure 4a shows the divergence plot for m 3, L = 27. is irrelevant in the calculation of the fractal dimen- We see that the plot consists of two parts, a dotted sion and estimation of the largest positive Lyapunov curve-like part and a heavy black part. Do they have exponent. Hence, the former should be removed as the same origin? Figure 5a shows the 4(k) curve, much as possible in these calculations. Since the which is by no means linear. Hence we should con- damaging effect ofthe tangential motion is enhanced dude that each part has its distinct origin, and that when & is decreased, too small a sampling time is the above assumption does not hold in this case. not recommended. Also &t is suggested not to exceed The answer is rather simple, however. There are the optimal delay time. certain small distances like dis(x~,x~-+~), with w very A note on the Lorenz system needs be made. This small, and their kth iterations can also be very small. system is very complicated in that, even when & is These points obviously correspond to the orbital not small (for example, 6t = 0.009), the tangential motion, and cannot be regarded as small distances motion still occupies a large fraction in the diverbetween orbits. Points in the divergence plot corre- gence plot, and the 4(k) curve is not linear by sponding to these points merely reflect changes of expression (1), if condition (3) is not imposed. This the phase velocity along the orbit, and such points was probably the reason that Wolf et al. [5] used a will increase if & is decreased. Let us call this part very large öt to estimate the largest Lyapunov exof the motion tangential motion. The tangential mo- ponent for this attractor. tion contributes a dimension nearly one, corre- Another important note should also be made. The sponding to a Lyapunov exponent equal to zero [191. optimal values of m and L & do not change with the Hence this motion should be excluded when calcu- sampling time &. Though condition (3) is suggested lating the fractal dimension and estimating the larg- when calculating the 4(k) curve to estimate the Lyaest Lyapunov exponent. A possible way of doing this punov exponent, it is not needed when reconstructis to add an additional condition to expression (1), ing a state space, since the tangential motion is an namely integral part of the motion on the attractor. j i~>w. (3) This work is partly supported by the National Ba- 157

6 Volume 181, number 2 PHYSICS LETTERS A 4 October 1993 sic Research Project Nonlinear Science. [7] A. Provenzale, L.A. Smith,R. Vio and G. Murante, Physica D58 (1992) 31. [8]A. Cenys and K. Pyragus, Phys. Lett. A 129 (1988) 227. References [9] Z. Aleksic, Physica D 52 (1991) 362. [10] W. Liebert, K. Pawelzik and H.G. Schuster, Europhys. Lett. 14(1991)521. [1] N.H. Packard, J.P. Crutchfield, J.D. Farmer and R.S. Shaw, [11] M.B. Kennel, R. Brown and H.D.I. Abarbanel, Phys. Rev. Phys. Rev. Lett. 45 (1980) 712; A45 (1992) F. Takens, in: Lecture notes in mathematics, Vol [12] J.M. Nese, Physica D 35 (1989) 237. Dynamical systemsand turbulence, eds. D.A. Rand and L.S. [13] H.D.I. Abarbenel, R. Brown and M.B. Kennel, J. Nonlinear Young(Springer,Berlin, 1981) p.366. Sci. 1(1991) 175;2 (1992) 343. [2] P. Grassberger and I. Procaccia, Physica D 9 (1983) 189; [14]A.M. Fraser and H.L. Swinney, Phys. Rev. A 33 (1986) Phys.Rev.A28 (1983) [3]J.-P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57 (1985) [15] M. Hénon, Commun. Math. Phys. 50 (1979) [16]O.E. Rossler, Phys. Lett. A 71(1979)155. [4]M. Sano and Y. Sawada, Phys. Rev. Lett. 55 (1985)1082. [17]E.N. Lorenz, J. Atmos. Sci. 20 (1963) 130. [5]A. Wolf, J.B. Swift, H.L. Swinneyand J.A. Vastano, Physica [18]M.C. Mackey and L. Glass, Science 197 (1977) 287. Dl6(1985)285. [l9]h.haken,phys.lett.a94(1983)71. [6]X. Zeng, R. Eykholt and R.A. Pieke, Phys. Rev. Lett. 66 [20]J. Theiler, Phys. Rev. A 34 (1986) 2427; J. Opt. Soc. Am. (1991) A7 (1990)