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Kato Chaos in Linear Dynamics

Author

Listed:
  • Lixin Jiao

    (School of Disciplinary Basics and Applied Statistics, Zhuhai College of Science and Technology (Zhuhai College of Jilin University), Zhuhai 519041, China
    Department of E-Business, South China University of Technology, Guangzhou 510006, China)

  • Lidong Wang

    (School of Disciplinary Basics and Applied Statistics, Zhuhai College of Science and Technology (Zhuhai College of Jilin University), Zhuhai 519041, China)

  • Heyong Wang

    (Department of E-Business, South China University of Technology, Guangzhou 510006, China)

Abstract

This paper introduces the concept of Kato chaos to linear dynamics and its induced dynamics. This paper investigates some properties of Kato chaos for a continuous linear operator T and its induced operators T ¯ . The main conclusions are as follows: (1) If a linear operator is accessible, then the collection of vectors whose orbit has a subsequence converging to zero is a residual set. (2) For a continuous linear operator defined on Fréchet space, Kato chaos is equivalent to dense Li–Yorke chaos. (3) Kato chaos is preserved under the iteration of linear operators. (4) A sufficient condition is obtained under which the Kato chaos for linear operator T and its induced operators T ¯ are equivalent. (5) A continuous linear operator is sensitive if and only if its inducing operator T ¯ is sensitive. It should be noted that this equivalence does not hold for nonlinear dynamics.

Suggested Citation

  • Lixin Jiao & Lidong Wang & Heyong Wang, 2023. "Kato Chaos in Linear Dynamics," Mathematics, MDPI, vol. 11(16), pages 1-9, August.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:16:p:3540-:d:1218430
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    References listed on IDEAS

    as
    1. Heng Liu & Fengchun Lei & Lidong Wang, 2013. "Li-Yorke Sensitivity of Set-Valued Discrete Systems," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-5, December.
    2. Félix Martínez-Giménez & Alfred Peris & Francisco Rodenas, 2021. "Chaos on Fuzzy Dynamical Systems," Mathematics, MDPI, vol. 9(20), pages 1-11, October.
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