Order patterns and chaos
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 , , and the Bernoulli shift , modelling the tossing of a fair coin, , , are isomorphic via the following recipe: if , set ; if , set (for any ‘typical’ initial point ). 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 , , and any binary block of length L, with , then there exists such that the symbolic sequence generated by the orbit segment is precisely . Here and below, and . Let us remind that the symbol corresponding to is 0 or 1 depending on whether or , respectively, the partition being a generating
Permutation entropy
The sets (each being, in general, a union of intervals) appear in the theory and practice of permutation entropy. Given a closed interval and a map with invariant measure μ (i.e., for every Borel set ), define the partition of I as and the topological permutation entropy of order as 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 is forbidden for f, then all the patterns with , are also forbidden for f. Denote now by 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 be a linearly ordered set by means of the order <, endowed with a metric , and let be a bijection (or, equivalently, an M-permutation). We define the discrete Lyapunov exponent (DLE) of f on , ,
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.
References (8)
- et al.
Physica D
(2005) - et al.
Iterated Maps on the Interval as Dynamical Systems
(1997) - et al.
Nonlinearity
(2002) Nonlinearity
(2003)
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