IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v32y2007i2p604-608.html
   My bibliography  Save this article

Individual chaos implies collective chaos for weakly mixing discrete dynamical systems

Author

Listed:
  • Liao, Gongfu
  • Ma, Xianfeng
  • Wang, Lidong

Abstract

Let X be a metric space, (X,f) a discrete dynamical system, where f:X→X is a continuous function. Let f¯ denote the natural extension of f to the space of all non-empty compact subsets of X endowed with Hausdorff metric induced by d. In this paper we investigate some dynamical properties of f and f¯. It is proved that f is weakly mixing (mixing) if and only if f¯ is weakly mixing (mixing, respectively). From this, we deduce that weak-mixing of f implies transitivity of f¯, further, if f is mixing or weakly mixing, then chaoticity of f (individual chaos) implies chaoticity of f¯ (collective chaos) and if X is a closed interval then f¯ is chaotic (in the sense of Devaney) if and only if f is weakly mixing.

Suggested Citation

  • Liao, Gongfu & Ma, Xianfeng & Wang, Lidong, 2007. "Individual chaos implies collective chaos for weakly mixing discrete dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 32(2), pages 604-608.
    • Handle: RePEc:eee:chsofr:v:32:y:2007:i:2:p:604-608
    DOI: 10.1016/j.chaos.2005.11.002
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077905010891
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2005.11.002?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. Fedeli, Alessandro, 2005. "On chaotic set-valued discrete dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 23(4), pages 1381-1384.
    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. Li, Risong, 2012. "A note on stronger forms of sensitivity for dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 45(6), pages 753-758.
    2. Deschrijver, Glad & O’Regan, Donal & Saadati, Reza & Mansour Vaezpour, S., 2009. "L-Fuzzy Euclidean normed spaces and compactness," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 40-45.
    3. Llorens-Fuster, Enrique & Petruşel, Adrian & Yao, Jen-Chih, 2009. "Iterated function systems and well-posedness," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1561-1568.
    4. 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.

    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. Jan Andres, 2020. "Multiple Periodic Solutions and Fractal Attractors of Differential Equations with n -Valued Impulses," Mathematics, MDPI, vol. 8(10), pages 1-21, October.
    2. Kwietniak, Dominik & Oprocha, Piotr, 2007. "Topological entropy and chaos for maps induced on hyperspaces," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 76-86.
    3. Andres, Jan, 2020. "Chaos for multivalued maps and induced hyperspace maps," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    4. 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.
    5. 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.
    6. 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.
    7. Peris, Alfredo, 2005. "Set-valued discrete chaos," Chaos, Solitons & Fractals, Elsevier, vol. 26(1), pages 19-23.
    8. Llorens-Fuster, Enrique & Petruşel, Adrian & Yao, Jen-Chih, 2009. "Iterated function systems and well-posedness," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1561-1568.
    9. Zhang, Gengrong & Zeng, Fanping & Liu, Xinhe, 2006. "Devaney’s chaotic on induced maps of hyperspace," Chaos, Solitons & Fractals, Elsevier, vol. 27(2), pages 471-475.
    10. Gu, Rongbao & Guo, Wenjing, 2006. "On mixing property in set-valued discrete systems," Chaos, Solitons & Fractals, Elsevier, vol. 28(3), pages 747-754.
    11. Román-Flores, Heriberto & Chalco-Cano, Y., 2005. "Robinson’s chaos in set-valued discrete systems," Chaos, Solitons & Fractals, Elsevier, vol. 25(1), pages 33-42.

    More about this item

    Statistics

    Access and download statistics

    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:32:y:2007:i:2:p:604-608. 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.