Article

# 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.
(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 parametric
unimodal maps. Furthermore, these PDFs depend on the value of the parameter,
what 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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• "Another option for approaching transient dynamics is to deal with coarse-grained versions of the associated time series and resort to the framework of applied symbolic theory. An efficient and accurate way to translate time series into symbolic representations is drawn by their ordinal patterns [67], which have been successfully applied to detect determinism [68], to the estimation of dynamical parameters [69], and to control chaotic systems [70]. Finally, on-line detection of events can be only performed if statistics are computed for short time series, and thus sliding windows must be used in order to meet this need. "
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• "As we have shown in [44], a chosen-plaintext attack on the cryptosystem defined in [35] can be used to obtain the symbolic sequence used in encryption. If the symbolic sequence was derived from the skew tent map, then the method described in [57] can be used to first determine the order patterns and second to estimate the control parameter. "
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