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Chaos in hyperspaces of nonautonomous discrete systems

Author

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  • Sánchez, Iván
  • Sanchis, Manuel
  • Villanueva, Hugo

Abstract

We study the interaction of some dynamical properties of a nonautonomous discrete dynamical system (X, f∞) and its induced nonautonomous discrete dynamical system (K(X),f∞¯), where K(X) is the hyperspace of non-empty compact sets in X, endowed with the Vietoris topology. We consider properties like transitivity, weakly mixing, points with dense orbit, density of periodic points, among others. We also present examples of nonautonomous discrete dynamical systems showing that transitivity, density of periodic points and sensitive dependence on initial conditions are independents on the unit interval, i.e., unlike autonomous discrete dynamical systems, in definition of Devaney chaotic there are not redundant conditions for NDS on the interval. Actually, our examples give an even more precise conclusion: the classical result stating that transitivity is a sufficient condition for an autonomous discrete dynamical system on the interval to be Devaney chaotic fails to be true for nonautonomous dynamical systems.

Suggested Citation

  • Sánchez, Iván & Sanchis, Manuel & Villanueva, Hugo, 2017. "Chaos in hyperspaces of nonautonomous discrete systems," Chaos, Solitons & Fractals, Elsevier, vol. 94(C), pages 68-74.
    • Handle: RePEc:eee:chsofr:v:94:y:2017:i:c:p:68-74
    DOI: 10.1016/j.chaos.2016.11.009
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    References listed on IDEAS

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    1. Peris, Alfredo, 2005. "Set-valued discrete chaos," Chaos, Solitons & Fractals, Elsevier, vol. 26(1), pages 19-23.
    2. 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. Daniel Jardón & Iván Sánchez & Manuel Sanchis, 2020. "Transitivity in Fuzzy Hyperspaces," Mathematics, MDPI, vol. 8(11), pages 1-9, October.
    2. Salman, Mohammad & Das, Ruchi, 2018. "Multi-sensitivity and other stronger forms of sensitivity in non-autonomous discrete systems," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 341-348.

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