CHAPTER IV CHARACTERIZATION OF WEAK AND STRONG CHAOS

Transcription

1 CHAPTER IV CHARACTERIZATION OF WEAK AND 4.1. INTRODUCTION STRONG CHAOS The essential characteristic property of chaotic motion is its sensi - tive dependence on initial conditions which leads to unpredictability of the long time evolution of chaotic systems. This is characterized by the positive Lyapunov exponent. As mentioned in chapter-i chaotic motion with small positive maximal Lyapunov exponent is termed as weak chaos. Chaotic solution with relatively large positive maximal Lyapunov exponent is called strong chaos. Classification of weak and strong chaos by positive maximal Lyapunov exponent is not of great use for experimental time series where calculation of maximal Lyapunov exponent is a time consuming process. Further, we need to know the maximal Lyapunov exponent for a range of values of a control parameter to identify weak and strong chaotic nature. Instead of maximal Lyapunov exponent, in the present chapter, we consider the k-step difference quantity of a state variable of DVP oscillator and study its characteristic property for weak and strong chaos and n-band chaos. 39

2 iuz Figure 4.1. Variation of maximal Lyapunov exponent A of DVP oscillator equation as a function of f in the interval (.112,.115).

3 CL. L) ) N) Lo.5 1. N >C).5. Interval t Figure 4.2. Numerically calculated P(/xk), k=51,52,53,54 for f= P is nonstationary.

4 4.2. CHARACTERIZATION OF WEAK AND STRONG CHAOS USING PROBABILITY DISTRIBUTION OF P(L\xk) Let {x}, (n = 1 1 2,.., n) be a set of values of the variable x of the DVP equation in the Poincaré map. We consider the k-step difference quantity Xm+k - X,n, (4.1) where m = 1,2,.., N', N' < N - k. We calculated the probability distribution of {LXk} as follows. The maximum and minimum values of { /.Xk} are determined. The range of LXk is divided into a finite number (M) of equal intervals. We then counted the number of occurrences of /xk in each interval and obtained the probability distribution P(Lxk) (here after called as P) by dividing the numbers by the total number Weak chaos Figure (4.1) shows the variation of maximal Lyapunov exponent A of DVP oscillator equation as a function of f in the interval (.112,.115). In this figure, there are many regions labelled by.j. in which A is positive but very small value indicating weak chaos. For example, for f = a chaotic motion with A =.56 is found. Figure (4.2) shows the evolution of P for k ,53,54. In the numerical calculation first 14 data are neglected as transient and next 2 x io data points are used. It is evident that P changes continuously 4

5 P--.5 IT to.25 MIM CM to Interval i Figure 4.3. Numerically calculated P(L\x k), k=51,52,53,54 for f=o.1u (onset of chaos). P is nonstationary.

6 L i iou Figure 4.4. X 2 (k, 1) versus k for f= P is nonstationary.

7 U Figure 4.5. X 2 (k, 1) versus k for f= P is nonstationary.

8 with k. That is, the distribution is nonstationary. Figure (4.3) shows the evolution of P for k=51,52,53,54 at 1= (onset of chaos. Here again the distribution is nonstationary. This is further verified by the chi-square, x2 test. The test quantity is defined as [18] x2(k;j) = M (R - S)2 i=1 (R + Si)' (4.2) where Ri and Si are the probabilities of the i-th interval for P(Lxk+J) and P(zx k) respectively. In the eq.(4.2) intervals with R = Si = are excluded. If two probabilities differ we get a large x2. For two similar distributions x2 will be small. The numerically calculated x2 for f = is plotted in fig.(4.4). The analysis is carried out for the values upto 5. The non-decreasing x2 implies the nonstationary characteristics of P. Thus, a complete probability distribution is impossible for weak chaos. Another example is shown in fig.(4.5) for f= (onset of chaos). A possible mechanism of nonstationary probability distribution can be a recurrence of memory loss and recovery of initial conditions [145,146]. The key to searching for memory recovery is the value of the Lyapunov exponent. Chern and Otsuka [152] applied information theory and a local Lyapunov exponent to characterize the locally deforming nature in chaos. Particularly, using self-information and mutual information flows they have shown that memory recovery is possible for chaos with a very small positive Lyapunov exponent. Thus, 41

9 .5 It LC).3 II LO x a LO.3 >< a.. Interval I Figure 4.6. Stationary evolution of P for f =.118 (one-band strong chaos).

10 L I Figure 4.7. Numerically calculated x 2 (lc, 1) at f=o.118 (strong chaos).

11 ct.1.5 [e1ii LO x 3- ci L() ci to.5..1 to ><.5 3- D. 2 4 Interval I Figure 4.8. Evolution of P for a two-band chaotic attractor. The value off is.115.

12 iuu 15 2 Figure 4.9. Numerically calculated 2 (k, 2) at f=o.115 (strong chaos). P is stationary.

13 the physical mechanism of the nonstationary probability distribution is the recurrence of memory loss and the recovery of initial conditions Strong chaos Stationary probability distribution is found for strong chaos. Figure (4.6) shows P for f.118. The Lyapunov exponent ) is.343. The calculated x 2 is plotted in fig.(4.7). After k = 25, x 2 becomes. That is, the distribution has evolved into a stationary state. This suggests that the variable x can be described as if it were generated by a random number generator with a certain probability distribution. A possible mechanism for stationary probability distribution is a complete loss of memory to the initial condition Two-band chaos We studied the distribution P for two-band chaos. A two-band chaotic attractor is found to occur for f.115. Figure (4.8) shows P for k 51, 52, 53, 54. A simple switching between two classes of probability distribution can be clearly seen. This is because x keeps on alternating between the two bands of the chaotic attractor. x2 (k, 1) and X 2 (k, 2) are calculated. X 2 (k, 1) is found to be large while X 2 (k, 2) becomes almost zero for large value of k (fig.4.9). The distribution is stationary. 42

14 Cl) C In (l) C In Figure 4.1. (a) S versus (f - f) and (b) S versus X in the in-in scale where f= We found S CX (f - f) where c=1.87 and S cx where a=2.17.

15 Degree of nonstationarity of P Nonstationary P occurs for a range of values of the control parameter for which maximal Lyapunov exponent is relatively small. We have investigated the dependence of the degree of nonstationarity of P with the control parameter, Lyapunov exponent and correlation dimension. For this purpose, we define the quantity S, the degree of nonstationarity of P(Lx k), as N S= urn 1E x2 (k;j), (4.3) N oo N k=1 where X2 (k; j) is given by eq.(4.2). For strong chaos x2 - for large k and hence S =. Since x2 does not decay to zero for weak chaos in the limit k p oc evidently S is non-zero. The magnitude of S characterizes the degree of different nonstationary probability distributions. S is calculated for a range of values of f in the DVP oscillator. Figure (4.lOa) shows S versus (f fe) in the ln - lu scale, where f = Circles represent the numerical data and continuous line is the best straight line fit. Power-law variation of S with (f - f) is observed. S is found to approach zero as (f - where a is a constant. We found S 5.771(f - f) 187. Figure (4.1b) depicts S versus A. We note that as A increases S decays to zero. That is, transition from nonstationary P to stationary P occurs as A increases from zero. S is found to scale with A as S x (f - f 217 x

16 4.3. CONCLUSION We have studied the characteristics of probability distribution P of chaotic attractors of the DVP equation. Stationary probability distribution is found for chaotic attractors with large positive Lyapunov exponent corresponding to strong chaos. Nonstationary probability distribution is found to occur for chaotic attractor with sufficiently small positive Lyapunov exponent which is related to weak chaos. The above study suggests that the analysis of probability distribution characteristics of chaos can be used to distinguish weak and strong chaos. 44