The decay of correlations in chaotic maps

Volume 111A, number 1,2

PHYSICS LETTERS

26 August 1985

T H E DECAY OF C O R R E L A T I O N S IN CHAOTIC MAPS Hans-Peter H E R Z E L and Wemer E B E L I N G Sektion Physik, Humboldt-Universit3t Berlin, Invalidenstr. 42, 1040 Berlin, GDR Received 22 May 1985; accepted for publication 10 June 1985

We investigate correlation functions for several chaotic maps. For the one-dimensional Belousov-Zhabotinsky map noise leads to a slower decay of correlations. In order to eliminate periodic components in a continuous three-dimensional model the second iterate of a next-amplitude map is used. Furthermore we discuss information-theoretical correlation measures, which describe the limited predictability of chaotic systems.

1. In recent years there has been considerable interest in discrete chaotic systems, which are related to several phenomena as turbulence and irregular behaviour in chemical reactions. Due to the trajectory instability associated with a positive Lyapunov exponent it appears desirable to use a statistical treatment [1-3]. The chaotic properties correspond to the asymptotic decay of correlations. The purpose of this work is to analyze this decay for the fully developed logistic map, a Belousov-Zhabotinsky model (BZ map), and for a continuous biochemical model. The autocorrelation function C(k) is defined by C ( k ) = ( x i , x i + k) = (xixi+ k) -
(1)

where the brackets stand for averaging over the stationary probability density P(x) in the case of the logistic map and over sufficiently long sequences (xi) in the case of our numerical computations. If the correlation fur/ction C(k) is a product of a modulation F(k) and an exponential decay [1-3] we can compare the corresponding exponent r from

C(k) = F(k) e x p ( - r k ) ,

(2)

with the Lyapunov exponent ?~calculated from the slope of the map. Both exponents are equal in the case of the map xi+ 1 = nxi (mod 1) (see ref. [2]); further computations in a continuous model confirm the idea that r = is not an exceptional case [5]. This comparison does not make sense for a power law decay of correlations 0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

[4] and for doubly symmetric maps [7]. 2. For the fully developed logistic map was found [11:

xi+ 1 = 4Xi(1 -- xi), X=ln2,

P(X) = It- 1 [x(1 - x ) ] - 1/2, (2)

C(k) = ~80k.

(3)

In spite of the high randomness of sequences (x i) generated by the map above, the x i are quite different from random numbers. This can be visualized by correlations of higher cumulants which do not decay to zero immediately. We obtained for example

=lA~8,

ifk=O,

= ~1,

i f k = 1,

= O,

ifk/>2,


(4)

ifk > log2m.

The corresponding integrals were computed by using the transformation z = (2/rr) arcsin X/~, which leads to the Bernoulli map Zi+l = 2zi (mod 1). Therefore the correlations of the function (2/rr) arcsin x/x decay exponentially [1]. <(2fir) arcsinx/~-/, (2/zr) arcsinvrxi+k > = ~ X 2 -k = h exp(-Mc).

(5)

Alternatively we may demonstrate correlation effects

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by disturbing the symmetry. Let us assume that x 0 < and introduce the symbol ( , ) standing for averaging over the left subinterval (0, ½). Using this special averaging procedure (initial condition) the correlations decay exponentially with K = 2X in the asymptotic regime (Xo,Xk) =4 -- l/rr2 , = {(1/2rr), =(I/2~r)[-1/(22k-

1)],

26 August 1985 C(K)/C(B)

A

B.4

~,

ifk = 0,

,'':

~

i f k = 1,

, /

'.A, A,,A,A',,A ~ I •

irks>2.

0"8[K

3. Low-dimensional chaotic systems are inevitably coupled to external fluctuations. Generalizations o f Lyapunov exponents in the presence o f noise lead to difficulties even in one-dimensional maps [ 10,12]. Therefore correlation functions could be an appropriate indicator for noise effects [13]. Noise-induced order which was found in a one-dimensional BZ map [8] can be clearly shown in terms o f correlations. The deterministic model which was developed to fit experimental data is written as follows: xi+ 1 = ( s i x i -- 411/3 + a) e x p ( - - x i ) + b ,

i f x i < 0.3,

xi+ 1 = c [ l O x i e x p ( - ~ x i ) ] 1 9

i f x i ~>0.3,

-0.4

""

Fig. 1. Normalized correlation function of the BZ map (b = 0.0232888279) without noise (full line) and with a noise level of a = 0.0008 (dashed line). 4. Next we aimed to compute a correlation function from a sequence o f maxima (Yi} which stems from a biochemical model [9]: dx/dt = 1 - B x - x y 2 - E x y + z , dy/dt = A ( x y 2 - y + O ) , dz/dt = F ( E x y - z ) .

(8)

Using a polynomial fit we found a Lyapunov exponent o f the map shown in fig. 3 ofX 1 = 0.31 + 0.02. The continuous correlation function ( y ( O ) , y ( t ) ) as well as (yi,Yi+k) have a periodic tail which corresponds to a two-band structure o f the map. Therefore we considered the decay o f correlations in one o f the bands: N

s =-1, = +1,

C(k) = N - 1 ~ Y 2 i Y 2 i + 2 k i=1

ifxi<4,

,r,

v.v v'.v,,

(6)

This result means that initial correlations due to nonstationary distributions decay more slowly than equilibrium correlations. Such an exponent ~ = 2X also has been derived in a quite different context [7].

+ b,

~

I~

(

N N - 1 G Y2i i=1

. (9)

ifx i >~,

a = 0.50607357,

c = 0.121205692.

(7)

In accordance with ref. [8] we consider b - 0.023288... and add to each x i a pseudorandom number equally distributed in the interval ( - a , o). As shown in fig. 1 noise leads to a slower decay o f correlations and a four-periodicity becomes visible. Fig. 2 exhibits how the correlators ( x 0 , x l > and (Xo,X 4) depend on the strength o f the perturbations. This result is consistent with previous calculations o f the Lyapunov exponent and the metric entropy [8].

C( ! ),'C(~,~) C (4 ],'C G,') ?,.6 % g,

~.a! ~.~

.

I

0.0~8~

g.~8

-0.2

Fig. 2. The correlators (xo,xl)/(Xo,Xo) (*) and (xo,x4)/
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t +1

2

F~"

.-~.

,\ 1

0

i.

".:.

I

0

I

I

I

1

I

,~.

I

I

I

,Yt 2

Fig. 3. Next-amplitude map (.4 = 4, B = 0.35, D = 0.1, E = 1.5, F = 0.2).

26 August 1985

All real measurements can be carried out with finite precision only. Naturally this leads to a measurement partition of the phase space into boxes which are labeled by indices i(i = 1..... M). Recording those partition elements visited by an orbit, the time evolution is translated into a sequence of symbols, a socalled information cylinder. The map itself may be considered as an information source. The boxes are weighted by their probability Pi which may be given in terms of their relative occurrence in a symbolic sequence. We definep!k) asthe probability for anelement of the sequence to fall into box i and for the element which follows k iterations later to fall into box/. We assume that we know the stationary probabilities for a single measurement Pi (i = 1..... M). Then the transinformation I(k) gives the average amount of information contained in a prediction corresponding to k iterations into the future, M i ( k ) = ~ p!k)ln(p}k)/pip])" 41=i

(10)

I(0) stands for the Shannon information Evidently fig. 4 shows a decay which is comparable with an exponential one: C(0) e x p ( - 2 X l k ) (dashed line). Such a computation of a correlation function can be applied easily to any experimental sequence of maxima, periods etc. and results in an estimation of the degree of randomness. 5. The transinformation I ( k ) is another correlation measure which describes the decay of correlations in the context of information sources [ 11,14,15 ].

M

i~__l Pi ln p-f 1 (oo) ~

and the mixing property leads to p!~ - PIP/, i.e. I ( ~ ) = O. Therefore the quantity C (k) = I(k) - I ( ~ ) may be used as a measure of correlations. In the special case of a Markov chain of first order the initial decay [I(0) - I ( 1 ) ] is equal to the metric entropy,

- H I =[(0) - I ( 1 ) M M

H=H 2 8.86

8.84

\

\

8.82

In the general case this simple relation does not hold [7,12,15] and the entropy of stationary and ergodic sequences is given by n = lim H , Ir,

8.88

-8.82

Fig. 4. The decay of correlation in one of the bands (N= 2000).

p - + oo

where H r are the higher-order Shannon entropies [ 16]. We note that for sources of mth order holds H = Hm +1 - Hm" By means of the source entropy the following correlation measure may be defined 3

Volume 111A, number 1,2

PHYSICS LETTERS

I(K) 3

0

I

I

I

I

I

2

4

6

fl

18

I K

12

Fig. 5. Transinformation for the map in fig. 3 using 4000 maxima

c"(e) = k- 1~k - H.

F o r proper stochastic sources this quantity is monotonously decreasing. It was shown [14] for the Bernoulli transformation xi+ 1 = 2xi (mod 1) that I(k) decreases at a linear rate, which for special partitions equals to the Lyapunov exponent X = In 2. In contrast to the autocorrelation function C(k) the transinformation does not decay immediately to zero for the fully developed logistic map indicating the deterministic character o f the sequences [ 14]. In the case o f the BZ map I(k) as well as C(k) (see fig. 1) decays more slowly due to small noise. Therefore we may state that the system is better predictable in the presence o f fluctuations. Finally we want to demonstrate the decay o f the transinformation for the next-amplitude map given in fig. 3. We use a rough partition into 20 boxes and find out that I(k) initially decreases faster than [I(0) 2Xlk ] (dashed line in fig. 5) and that I(k) reaches -

26 August 1985

In 2 = 1 bit. This information is preserved due to the two-band structure o f the map. If the bands merge due to additive noise then I(k) decays to zero [ 10]. Summarizing we are able to say that Lyapunov exponents and dimensions are o f great importance to make chaos and noise distinguishable. But if the chaotic character o f a dynamical system is obviously clear, correlation functions and the transinformation are important measures which are easily computed. They are useful tools for analyzing the predictability, to study the influence o f noise and may even give sometimes a rough estimation of a positive Lyapunov exponent.

References [1] S. Grossmann and S. Thomae, Z. Naturforseh. 32a (1977) 1353. [2] R.L. Stratonovich, Dokl. Akad. Nauk 267 (1982) 355. [3] T. Nagashima and H. Haken, Phys. Lett. 96A (1983) 385. [4] S. Grossmann, Lecture at the Leopoldina-Seminar on Phase transitions (Halle, 1985). [5] W.S. Anishenko, Contribution at the 3rd. Conf. on Irreversible processes and dissipative structures (Kiihlungsborn, 1985). [6] A. Arn6odo and D. Sornette, Phys. Rev. Lett. 52 (1984) 1857. [7] G. GySrgyi and P. Szepfalusy, preprint (1985). [8] K. Matsumoto and I. Tsuda, J. Stat. Phys. 31 (1983) 87. [9] Th. Schulmeister, Stud. Biophys. 72 (1978) 205. [10] H. Herzel, W. Ebeling and Th. Scbulmeister, in: Proc. on System analysis and simulation (Berlin, 1985). [11] J.D. Farmer, Z. Naturforsch. 37a(1982) 1304. [ 12 ] J. Crutchfield and N.H. Packard, Physica 7D ( 1983) 201. [13] W. Ebeling and Yu.L. Klimontovich, Selforganization and turbulence (Teubner, Leipzig, 1984). [14] R.W. Leven and B. Pompe, Ann. Phys~ (Leipzig), to be published. [15] B. Pompe and R.W. Leven, submitted to Physica D. [16] W. Ebeling and R. Feistel, Physik der Selbstorganisation und Evolution (Akademie-Verlag, Berlin, 1985).