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Chaos on Fuzzy Dynamical Systems

Author

Listed:
  • Félix Martínez-Giménez

    (Institut Universitari de Matemàtica Pura i Aplicada, Universitat Politècnica de València, 46022 València, Spain)

  • Alfred Peris

    (Institut Universitari de Matemàtica Pura i Aplicada, Universitat Politècnica de València, 46022 València, Spain)

  • Francisco Rodenas

    (Institut Universitari de Matemàtica Pura i Aplicada, Universitat Politècnica de València, 46022 València, Spain)

Abstract

Given a continuous map f : X → X on a metric space, it induces the maps f ¯ : K ( X ) → K ( X ) , on the hyperspace of nonempty compact subspaces of X , and f ^ : F ( X ) → F ( X ) , on the space of normal fuzzy sets, consisting of the upper semicontinuous functions u : X → [ 0 , 1 ] with compact support. Each of these spaces can be endowed with a respective metric. In this work, we studied the relationships among the dynamical systems ( X , f ) , ( K ( X ) , f ¯ ) , and ( F ( X ) , f ^ ) . In particular, we considered several dynamical properties related to chaos: Devaney chaos, A -transitivity, Li–Yorke chaos, and distributional chaos, extending some results in work by Jardón, Sánchez and Sanchis (Mathematics 2020, 8, 1862) and work by Bernardes, Peris and Rodenas (Integr. Equ. Oper. Theory 2017, 88, 451–463). Especial attention is given to the dynamics of (continuous and linear) operators on metrizable topological vector spaces (linear dynamics).

Suggested Citation

  • Félix Martínez-Giménez & Alfred Peris & Francisco Rodenas, 2021. "Chaos on Fuzzy Dynamical Systems," Mathematics, MDPI, vol. 9(20), pages 1-11, October.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:20:p:2629-:d:659025
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    References listed on IDEAS

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    1. Daniel Jardón & Iván Sánchez & Manuel Sanchis, 2020. "Transitivity in Fuzzy Hyperspaces," Mathematics, MDPI, vol. 8(11), pages 1-9, October.
    2. Peris, Alfredo, 2005. "Set-valued discrete chaos," Chaos, Solitons & Fractals, Elsevier, vol. 26(1), pages 19-23.
    3. Fu, Heman & Xing, Zhitao, 2012. "Mixing properties of set-valued maps on hyperspaces via Furstenberg families," Chaos, Solitons & Fractals, Elsevier, vol. 45(4), pages 439-443.
    4. Banks, John, 2005. "Chaos for induced hyperspace maps," Chaos, Solitons & Fractals, Elsevier, vol. 25(3), pages 681-685.
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    Cited by:

    1. Lixin Jiao & Lidong Wang & Heyong Wang, 2023. "Kato Chaos in Linear Dynamics," Mathematics, MDPI, vol. 11(16), pages 1-9, August.

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