Abstract
This article deals with the mathematical modeling of Tsallis entropy in fuzzy dynamical systems. At first, the concepts of Tsallis entropy and Tsallis conditional entropy of order where is a positive real number not equal to 1, of fuzzy partitions are introduced and their mathematical behavior is described. As an important result, we showed that the Tsallis entropy of fuzzy partitions of order satisfies the property of sub-additivity. This property permits the definition of the Tsallis entropy of order of a fuzzy dynamical system. It was shown that Tsallis entropy is an invariant under isomorphisms of fuzzy dynamical systems; thus, we acquired a tool for distinguishing some non-isomorphic fuzzy dynamical systems. Finally, we formulated a version of the Kolmogorov–Sinai theorem on generators for the case of the Tsallis entropy of a fuzzy dynamical system. The obtained results extend the results provided by Markechová and Riečan in Entropy, 2016, 18, 157, which are particularized to the case of logical entropy.
Highlights
The concept of entropy to ergodic theory was introduced by Kolmogorov [1] and Sinai [2] in relation to the problem of isomorphisms of dynamical systems
In our work [9], we introduced the concept of logical entropy of fuzzy dynamical systems
The main objective of this article was to construct a mathematical model for the Tsallis entropy of fuzzy dynamical systems
Summary
The concept of entropy to ergodic theory was introduced by Kolmogorov [1] and Sinai [2] in relation to the problem of isomorphisms of dynamical systems. The logical entropy of partitions in a product MV-algebra (cf [53,54,55]) was defined and studied in Reference [10] Another recently published paper [56] was devoted to the mathematical modeling of Tsallis entropy in product MV-algebra dynamical systems. It is shown that the proposed definitions of Tsallis entropies generalize the logical entropy of fuzzy partitions studied in Reference [9]; it is enough to put q = 2. The following definition of logical entropy and conditional logical entropy of fuzzy partitions was introduced in Reference [9]. The definition of the Shannon-type entropy of fuzzy partitions was proposed in Reference [29], and is given as follows.
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