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Introduction and main concepts
Published in Marcio Eisencraft, Romis Attux, Ricardo Suyama, Chaotic Signals in Digital Communications, 2018
A similar procedure can be used to control a system trajectory so that a message can be conveniently codified on it. A chaotic signal is characterized by an aperiodic and apparently random waveform that produces a sequence of maxima and minima [2]. One can associate the digital symbol 1 each time the wave overpasses a previously specified value, while the symbol 0 is associated with signal excursions that underpass another prior established value [3,4,28]. By doing so, one can view a chaotic system as a binary sequence generator. Furthermore, this generator can be controlled by using small perturbations in a framework of an implemented control of chaos strategy [9, 10, 17]. Thus, the chaotic dynamics can be controlled to produce desired digital sequences. The control of chaos methodology used to accomplish the symbolic codification is named feedback control of symbolic dynamics. Let us discuss how it works.
Semantic Information Extraction
Published in S. Sitharama Iyengar, Richard R. Brooks, Distributed Sensor Networks, 2016
In symbolic dynamics, the numeric time series associated with a system’s dynamics are converted into streams of symbols. The streams define a formal language where any substring in the stream belongs to the language. Conversions of physical measurements to symbolic dynamics and the analysis of the resulting strings of symbols have been used for characterizing nonlinear dynamical systems as they simplify data handling while retaining important qualitative phenomena. This also allows usage of complexity measures defined on formal languages made of symbol strings to characterize the system dynamics [Kurths 95]. The distance between individual symbols is not defined so there is no notion of linearity.
Historical Development of HRV Analysis
Published in Herbert F. Jelinek, David J. Cornforth, Ahsan H. Khandoker, ECG Time Series Variability Analysis, 2017
Kurths and Voss introduced symbolic dynamics into HRV analysis (Kurths et al. 1995; Voss et al. 1996b) by developing special optimized measures for the analysis of HR dynamics. The application of symbolic dynamics has been proven to be sufficient for the investigation of complex systems and describes dynamic aspects within time series (Voss et al. 1996b). The concept of symbolic dynamics is based on a coarse-graining of the dynamics of the original HR time series applying a defined number of symbols. To classify dynamic changes within the NN interval time series, the NN intervals are first transformed into a symbol sequence with symbols from a given alphabet A={0,1,2,3}. Thus, 64 different word types using three successive symbols from the alphabet to characterize symbol strings are obtained. The resulting histogram contains the distribution of each single word within a word sequence. Based on the probability distribution of each word type, several indices can be calculated: the Shannon and Renyi entropy of the word distribution, a complexity measure; the number of seldom (p<0.001) or never occurring word types referred to as forbidden words; wpsum02, the relative portion of words consisting only of the symbols “0” and “2” (measure for decreased HRV); wpsum13, the relative portion of words consisting only of the symbols “1” and “3” (measure for increased HRV); wsdvar, the standard deviation of a word sequence; phvarX, the portion of high-variability patterns in the NN interval time series >X ms; and plvarX, the portion of low-variability patterns in the NN interval time series <X ms.
Induced hyperspace dynamical systems of symbolic dynamical systems
Published in International Journal of General Systems, 2018
Zhiming Li, Minghan Wang, Guo Wei
In mathematics, symbolic dynamics is the practice of modeling a topological or smooth dynamical system by a discrete space consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics (evolution) given by the shift operator. Symbolic systems have various representative and complicated dynamical properties (Robinson 1999; Bowen 2008; Kari 2012; Pivato 2012; Li 2016). However, to the best of our knowledge, there is less attention paid to the induced hyperspace dynamics of symbolic systems in the literature (Wang and Wei 2008). The purpose of this paper is to investigate the topological entropy, expansivity of the induced hyperspace dynamical systems of symbolic dynamical systems.