Estimation of the control parameter from symbolic sequences: unimodal maps with variable critical point.

Instituto de Fisica Aplicada, Consejo Superior de Investigaciones Cientificas, Serrano 144, 28006 Madrid, Spain.
Chaos (Woodbury, N.Y.) (Impact Factor: 1.76). 07/2009; 19(2):023125. DOI: 10.1063/1.3155072
Source: PubMed

ABSTRACT The work described in this paper can be interpreted as an application of the order patterns of symbolic dynamics when dealing with unimodal maps. Specifically, it is shown how Gray codes can be used to estimate the probability distribution functions (PDFs) of the order patterns of unimodal maps whose dynamics is controlled by an external parameter. Furthermore, these PDFs depend on the value of the external parameter, which eventually provides a handle to estimate the parameter value from symbolic sequences (in form of Gray codes), even when the critical point depends on the parameter.

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    ABSTRACT: In this paper we provide a closed mathematical formulation of our previous results in the field of symbolic dynamics of unimodal maps. This being the case, we discuss the classical theory of applied symbolic dynamics for unimodal maps and its reinterpretation using Gray codes. This connection was previously emphasized but no explicit mathematical proof was provided. The work described in this paper not only contributes to the integration of the different interpretations of symbolic dynamics of unimodal maps, it also points out some inaccuracies that exist in previous works.
    Communications in Nonlinear Science and Numerical Simulation 07/2014; 19(7):2345–2353. DOI:10.1016/j.cnsns.2013.11.005 · 2.57 Impact Factor
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    ABSTRACT: This book describes a study of permutation complexity, a subject that might be thought of as a kind of symbolic dynamics where the basic elements are ordinal patterns. Ordinal patterns are permutations defined by the order relations among points in the orbits of dynamical systems. The primary topics of the book are ordinal patterns, permutations, entropy and complexity. The book is aimed primarily at researchers in nonlinear dynamics and complex systems, but it is also accessible to graduate students. The common element among the topics studied here is a hypothetical order structure on the state space of a dynamical system substantiated in the form of ordinal patterns. The state space must be totally ordered, but is otherwise arbitrary. Thus it includes discrete sets and n-dimensional intervals, for example. The class of dynamical systems that are studied include stochastic sytems. Under some mild assumptions it can be shown that some ordinal patterns cannot be realized, contrary to the situation with symbol patterns. The existence of forbidden ordinal patterns, it turns out, can be used as a finger print of determinist orbit generation since, with probability one, random dynamics has no forbidden order patterns. The first three chapters of the book introduce the main concepts, show how they can be linked, and describe some applications of ordinal analysis. Mappings and their relationship with the structure of their admissible and forbidden patterns are taken up in the next two chapters. Here there is also a detailed study of the shift and signed-shift transformations. The remaining chapters discuss permutation entropy (metric and topological), discrete entropy, the detection of determinacy, and analysis of cellular automata and coupled map lattice systems.

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