Abstract
In the present chapter, we made a detailed analysis of the different regimes of certain chaotic systems and their correspondence with the change in the normalized Shannon entropy, Statistical Complexity, and Fisher information measure. We construct a bidimensional plane composed of the selection of a pair of the informational tools mentioned above (a casual plane is defined), in which the different dynamical regimes appeared very clear and give more information of the underlying process. In such a way, a plane composed of the normalized Shannon entropy, statistical complexity, normalized Shannon entropy, and Fisher information measure can be applied to follow the changes in the behavior variations of the nonlinear systems.
Highlights
In the space of few decades, chaos theory has jumped from the scientific literature into the popular realm, being regarded as a new way of looking at complex systems like brains or ecosystems
The mathematical tools applied once the probability distribution function (PDF) is available receive the name of informational tools; more precisely information theory quantifiers [5], the main feature of the quantifiers is exactly quantifying the amount of information coming from the time series (TS), originating in the dynamical system
Some ambiguities arise in the case in which one wishes to employ the Bandt and Pompe (BP)-PDF to construct local quantifiers
Summary
In the space of few decades, chaos theory has jumped from the scientific literature into the popular realm, being regarded as a new way of looking at complex systems like brains or ecosystems. R, the points will be apart from each other, determining a nonzero distance between them This fact could be interpreted by a certain kind of instability which reveals some information about the phase space population that was not available at earlier times [4]. The existence of simple “calibrated” sources such as the logistic map provides a means for a precise evaluation of the performance of these information quantifiers In this communication we take advantage such fact in order to show that planar representations constructed with two information theory-based quantifiers offer one possibility of visualizing many interesting details of chaos characteristics, including the fine structure of chaotic attractors. We exemplified their use showing the result on two chaotic maps: the logistic map and the delayed logistic map
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