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Tsallis Entropy of Fuzzy Dynamical Systems

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  • Dagmar Markechová

    (Department of Mathematics, Faculty of Natural Sciences, Constantine the Philosopher University in Nitra, A. Hlinku 1, SK-949 01 Nitra, Slovakia)

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 q , where q 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 q > 1 satisfies the property of sub-additivity. This property permits the definition of the Tsallis entropy of order q > 1 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.

Suggested Citation

  • Dagmar Markechová, 2018. "Tsallis Entropy of Fuzzy Dynamical Systems," Mathematics, MDPI, vol. 6(11), pages 1-19, November.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:11:p:264-:d:183710
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    References listed on IDEAS

    as
    1. Naudts, Jan, 2002. "Deformed exponentials and logarithms in generalized thermostatistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 316(1), pages 323-334.
    2. Constantino Tsallis & Celia Anteneodo & Lisa Borland & Roberto Osorio, 2003. "Nonextensive statistical mechanics and economics," Papers cond-mat/0301307, arXiv.org.
    3. Rosso, O.A & Martin, M.T & Plastino, A, 2003. "Brain electrical activity analysis using wavelet-based informational tools (II): Tsallis non-extensivity and complexity measures," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 320(C), pages 497-511.
    4. Hanel, Rudolf & Thurner, Stefan, 2007. "Generalized Boltzmann factors and the maximum entropy principle: Entropies for complex systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 380(C), pages 109-114.
    5. Almeida, M.P., 2001. "Generalized entropies from first principles," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 300(3), pages 424-432.
    6. Tong, S. & Bezerianos, A. & Paul, J. & Zhu, Y. & Thakor, N., 2002. "Nonextensive entropy measure of EEG following brain injury from cardiac arrest," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 305(3), pages 619-628.
    7. Tsallis, Constantino & Anteneodo, Celia & Borland, Lisa & Osorio, Roberto, 2003. "Nonextensive statistical mechanics and economics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 324(1), pages 89-100.
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