IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v94y2017icp68-74.html
   My bibliography  Save this article

Chaos in hyperspaces of nonautonomous discrete systems

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077916303368
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2016.11.009?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. Daniel Jardón & Iván Sánchez & Manuel Sanchis, 2020. "Transitivity in Fuzzy Hyperspaces," Mathematics, MDPI, vol. 8(11), pages 1-9, October.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Cánovas Peña, Jose S. & López, Gabriel Soler, 2006. "Topological entropy for induced hyperspace maps," Chaos, Solitons & Fractals, Elsevier, vol. 28(4), pages 979-982.
    2. 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.
    3. Daniel Jardón & Iván Sánchez & Manuel Sanchis, 2020. "Transitivity in Fuzzy Hyperspaces," Mathematics, MDPI, vol. 8(11), pages 1-9, October.
    4. Félix Martínez-Giménez & Alfred Peris & Francisco Rodenas, 2021. "Chaos on Fuzzy Dynamical Systems," Mathematics, MDPI, vol. 9(20), pages 1-11, October.
    5. Kwietniak, Dominik & Oprocha, Piotr, 2007. "Topological entropy and chaos for maps induced on hyperspaces," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 76-86.
    6. Wang, Yangeng & Wei, Guo & Campbell, William H. & Bourquin, Steven, 2009. "A framework of induced hyperspace dynamical systems equipped with the hit-or-miss topology," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1708-1717.
    7. Liu, Heng & Liao, Gongfu & Hou, Bingzhe, 2009. "The set-valued mapping induced by a non-minimal transitive system is Li–Yorke chaotic," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 826-830.
    8. Hou, Bingzhe & Liao, Gongfu & Liu, Heng, 2008. "Sensitivity for set-valued maps induced by M-systems," Chaos, Solitons & Fractals, Elsevier, vol. 38(4), pages 1075-1080.
    9. Román-Flores, H. & Chalco-Cano, Y., 2008. "Some chaotic properties of Zadeh’s extensions," Chaos, Solitons & Fractals, Elsevier, vol. 35(3), pages 452-459.
    10. Andres, Jan, 2020. "Chaos for multivalued maps and induced hyperspace maps," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    11. Daghar, Aymen & Naghmouchi, Issam, 2022. "Entropy of induced maps of regular curves homeomorphisms," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    12. Román-Flores, H. & Chalco-Cano, Y. & Silva, G.N. & Kupka, Jiří, 2011. "On turbulent, erratic and other dynamical properties of Zadeh’s extensions," Chaos, Solitons & Fractals, Elsevier, vol. 44(11), pages 990-994.
    13. Li, Risong, 2012. "A note on stronger forms of sensitivity for dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 45(6), pages 753-758.
    14. Camargo, Javier & Rincón, Michael & Uzcátegui, Carlos, 2019. "Equicontinuity of maps on dendrites," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 1-6.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:94:y:2017:i:c:p:68-74. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.