Elsevier

Physics Letters A

Volume 355, Issue 1, 19 June 2006, Pages 27-31
Physics Letters A

Order patterns and chaos

https://doi.org/10.1016/j.physleta.2006.01.093Get rights and content

Abstract

Chaotic maps can mimic random behavior in a quite impressive way. In particular, those possessing a generating partition can produce any symbolic sequence by properly choosing the initial state. We study in this Letter the ability of chaotic maps to generate order patterns and come to the conclusion that their performance in this respect falls short of expectations. This result reveals some basic limitation of a deterministic dynamic as compared to a random one. This being the case, we propose a non-statistical test based on ‘forbidden’ order patterns to discriminate chaotic from truly random time series with, in principle, arbitrarily high probability. Some relations with discrete chaos and chaotic cryptography are also discussed.

Introduction

Random systems and chaotic systems share some important features, both from the theoretical and practical point of view. Thus one can define truly random symbolic dynamics by means of chaotic maps, what boils down to the fact that the dynamical systems defined by the iteration of such maps are isomorphic (or conjugate) to shift systems on sequence spaces—standard models for stationary random processes—despite their different nature. For instance, the logistic map f(x)=4x(1x), 0x1, and the Bernoulli shift B(12,12), modelling the tossing of a fair coin, Xn:{head, tail}{0,1}, n=0,1,, are isomorphic via the following recipe: if fn(x0)[0,12), set Xn=0; if fn(x0)[12,1], set Xn=1 (for any ‘typical’ initial point x0). This and similar properties are exploited e.g. in the generation of pseudo-random sequences in different applications.

Although we will address the isomorphy between random and chaotic systems with more detail below, the actual focus of this Letter is rather on the differences between random and chaotic systems and, specifically, on the order relations (or order patterns) defined by chaotic orbits of piecewise monotone interval maps. Indeed, if an order pattern is missing, then its absence pervades all longer patterns in form of more missing order patterns. In other words, chaotic trajectory points, as random as they may look, cannot be ordered in arbitrary ways—in contrast to the orbits of random processes with arbitrary alphabets. Not occurring order patterns will be called forbidden patterns and (somewhat paradoxically) their ‘existence’ can be used to tell chaotic from random time series with, in principle, arbitrarily high probability. Furthermore, this method is also robust against noisy data and, under circumstances, it can be a practical alternative to more conventional techniques.

We will also refer to some relations to discrete chaos and chaotic cryptography. In fact, it was in the framework of discrete chaos and its applications to cryptography where the authors first noticed that determinism imposes some limitations on the permutations (i.e., order patterns) that a chaotic map can define directly by means of their orbits. Although the possible consequences for chaotic cryptography (and, eventually, for other application areas) are more of a theoretical sort, it seems nevertheless that there some basic limitations exist as for what can be done by means of chaotic maps if used in a straightforward, naive way.

In the last section we will come back to the relation between chaotic and symbolic dynamics since the different performance of deterministic and random systems, as measured by the order patterns, may seem at odds with the possibility of being isomorphic.

Section snippets

Order patterns

It is well known [1] that given, say, the logistic map f(x)=4x(1x), 0x1, and any binary block of length L, b1L=b1bL with bi{0,1}, then there exists x0[0,1] such that the symbolic sequence generated by the orbit segment {x0,f(x0),,fL1(x0)} is precisely b1L. Here and below, fn(x):=f(fn1(x)) and f0(x):=x. Let us remind that the symbol corresponding to fk(x0) is 0 or 1 depending on whether fk(x0)[0,1/2) or fk(x0)[1/2,1], respectively, the partition {[0,1/2),[1/2,1]} being a generating

Permutation entropy

The sets Pπ (each being, in general, a union of intervals) appear in the theory and practice of permutation entropy. Given a closed interval IR and a map f:II with invariant measure μ (i.e., μ(f−1B)=μ(B) for every Borel set BI), define the partition PL* of I asPL*={Pπ:πσL} and the topological permutation entropy of order L2 asH¯0(L)(f)=1L1log|PL*|, where || denotes here cardinality. If f is a piecewise monotone interval map (i.e., there is a finite partition of I into intervals, such

Forbidden patterns

The bottom line of Proposition 1 is that, for every piecewise monotone interval map f, there are order patterns of minimal length which cannot occur in any orbit. We will call them forbidden patterns for f and recall how their absence paradoxically pervades all longer patterns in form of forbidden outgrowth patterns: If πforb=[π1,,πL0] is forbidden for f, then all the patterns [*,π1+n,*,,*,πL0+n,*]σL with n=0,1,,LL0, are also forbidden for f. Denote now by σLout(πforb) the family of length

Discrete Lyapunov exponent

Interestingly enough, the authors came across the fore-going questions when developing the theory of discrete chaos [5], [6] and, specifically, when generalizing the concept of Lyapunov exponent to maps on finite sets—a concept we call discrete Lyapunov exponent.

Definition 1

Let S={s0,s1,,sM1} be a linearly ordered set by means of the order <, endowed with a metric d(,), and let F:SS be a bijection (or, equivalently, an M-permutation). We define the discrete Lyapunov exponent (DLE) of f on (S,<,d), λF,

Chaos and symbolic dynamics

To close our excursion through order, chaos and randomness, we would like to return to one of the most intriguing aspects: the isomorphy of random and chaotic systems, despite the different quality of their orbits.

Any stationary stochastic process corresponds to a measure-preserving shift transformation on a sequence space in a standard way [3], [8]. Such shift systems, sometimes called sequence space models, allow to focus on the random process itself as given by the probability distribution

Conclusion

Chaos manages easily to reproduce an exponentially growing manifold of patterns (like symbol blocks) but, subject to very mild mathematical conditions, cannot cope with a super-exponentially growing manifold such as that of order patterns. This shortcoming has been exposed by means of the permutation entropy and the discrete Lyapunov exponent. Only truly random dynamical systems (i.e., stationary random processes with arbitrary alphabets) are up to the task. A first consequence of this

Acknowledgements

J.M.A. has been partially supported by the Spanish Ministry of Education and Science, grant GRUPOS 04/79. L.K. has been supported by the Spanish Ministry of Education and Science, grant SAB2004-0048. L.K. also thanks NSF for partial support.

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