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Critical homoclinic orbits lead to snap-back repellers

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  • Gardini, Laura
  • Sushko, Iryna
  • Avrutin, Viktor
  • Schanz, Michael

Abstract

When nondegenerate homoclinic orbits to an expanding fixed point of a map f:X→X,X⊆Rn, exist, the point is called a snap-back repeller. It is known that the relevance of a snap-back repeller (in its original definition) is due to the fact that it implies the existence of an invariant set on which the map is chaotic. However, when does the first homoclinic orbit appear? When can other homoclinic explosions, i.e., appearance of infinitely many new homoclinic orbits, occur? As noticed by many authors, these problems are still open. In this work we characterize these bifurcations, for any kind of map, smooth or piecewise smooth, continuous or discontinuous, defined in a bounded or unbounded closed set. We define a noncritical homoclinic orbit and a homoclinic orbit of an expanding fixed point is structurally stable iff it is noncritical. That is, only critical homoclinic orbits are responsible for the homoclinic explosions. The possible kinds of critical homoclinic orbits will be also investigated, as well as their dynamic role.

Suggested Citation

  • Gardini, Laura & Sushko, Iryna & Avrutin, Viktor & Schanz, Michael, 2011. "Critical homoclinic orbits lead to snap-back repellers," Chaos, Solitons & Fractals, Elsevier, vol. 44(6), pages 433-449.
    • Handle: RePEc:eee:chsofr:v:44:y:2011:i:6:p:433-449
    DOI: 10.1016/j.chaos.2011.03.004
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    References listed on IDEAS

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    1. Marotto, F.R., 2005. "On redefining a snap-back repeller," Chaos, Solitons & Fractals, Elsevier, vol. 25(1), pages 25-28.
    2. Shi, Yuming & Yu, Pei, 2006. "Study on chaos induced by turbulent maps in noncompact sets," Chaos, Solitons & Fractals, Elsevier, vol. 28(5), pages 1165-1180.
    3. Gao, Yinghui, 2009. "Complex dynamics in a two-dimensional noninvertible map," Chaos, Solitons & Fractals, Elsevier, vol. 39(4), pages 1798-1810.
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    Cited by:

    1. Khellat, Farhad & Ghaderi, Akashe & Vasegh, Nastaran, 2011. "Li–Yorke chaos and synchronous chaos in a globally nonlocal coupled map lattice," Chaos, Solitons & Fractals, Elsevier, vol. 44(11), pages 934-939.
    2. Stankevich, N.V. & Gonchenko, A.S. & Popova, E.S. & Gonchenko, S.V., 2023. "Complex dynamics of the simplest neuron model: Singular chaotic Shilnikov attractor as specific oscillatory neuron activity," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    3. Commendatore, Pasquale & Pinto, Antonio & Sushko, Iryna, 2014. "A post-Keynesian model of growth and distribution with a constraint on investment," Structural Change and Economic Dynamics, Elsevier, vol. 28(C), pages 12-24.
    4. Laura Gardini & Roya Makrooni & Iryna Sushko, 2016. "Cascades of Alternating Smooth Bifurcations and Border Collision Bifurcations in a Family of Discontinuous Linear-Power Maps," Gecomplexity Discussion Paper Series 201603, Action IS1104 "The EU in the new complex geography of economic systems: models, tools and policy evaluation", revised Mar 2016.
    5. Ingrid Kubin & Laura Gardini, 2022. "On the significance of borders: the emergence of endogenous dynamics," Journal of Economic Interaction and Coordination, Springer;Society for Economic Science with Heterogeneous Interacting Agents, vol. 17(1), pages 41-62, January.
    6. Laura Gardini & Iryna Sushko, 2018. "Growing through chaos in the Matsuyama map via subcritical flip and bistability," Working Papers 1801, University of Urbino Carlo Bo, Department of Economics, Society & Politics - Scientific Committee - L. Stefanini & G. Travaglini, revised 2018.
    7. Matsuo, Akihito & Asahara, Hiroyuki & Kousaka, Takuji, 2012. "Bifurcation structure of chaotic attractor in switched dynamical systems with spike noise," Chaos, Solitons & Fractals, Elsevier, vol. 45(6), pages 795-804.
    8. Roya Makrooni & Laura Gardini, 2015. "Bifurcation structures in a family of one-dimensional linear-power discontinuous maps," Gecomplexity Discussion Paper Series 7, Action IS1104 "The EU in the new complex geography of economic systems: models, tools and policy evaluation", revised Jan 2015.
    9. Gardini, Laura & Sushko, Iryna, 2019. "Growing through chaos in the Matsuyama map via subcritical flip bifurcation and bistability," Chaos, Solitons & Fractals, Elsevier, vol. 124(C), pages 52-67.

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