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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