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Bifurcation structure of parameter plane for a family of unimodal piecewise smooth maps: Border-collision bifurcation curves


  • Sushko, Iryna
  • Agliari, Anna
  • Gardini, Laura


We study the structure of the 2D bifurcation diagram for a two-parameter family of piecewise smooth unimodal maps f with one break point. Analysing the parameters of the normal form for the border-collision bifurcation of an attracting n-cycle of the map f, we describe the possible kinds of dynamics associated with such a bifurcation. Emergence and role of border-collision bifurcation curves in the 2D bifurcation plane are studied. Particular attention is paid also to the curves of homoclinic bifurcations giving rise to the band merging of pieces of cyclic chaotic intervals.

Suggested Citation

  • Sushko, Iryna & Agliari, Anna & Gardini, Laura, 2006. "Bifurcation structure of parameter plane for a family of unimodal piecewise smooth maps: Border-collision bifurcation curves," Chaos, Solitons & Fractals, Elsevier, vol. 29(3), pages 756-770.
  • Handle: RePEc:eee:chsofr:v:29:y:2006:i:3:p:756-770
    DOI: 10.1016/j.chaos.2005.08.107

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    References listed on IDEAS

    1. Day, Richard H, 1982. "Irregular Growth Cycles," American Economic Review, American Economic Association, vol. 72(3), pages 406-414, June.
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    Cited by:

    1. Radi, Davide & Gardini, Laura & Avrutin, Viktor, 2014. "The role of constraints in a segregation model: The symmetric case," Chaos, Solitons & Fractals, Elsevier, vol. 66(C), pages 103-119.
    2. Sushko, Iryna & Gardini, Laura & Matsuyama, Kiminori, 2014. "Superstable credit cycles and U-sequence," Chaos, Solitons & Fractals, Elsevier, vol. 59(C), pages 13-27.
    3. Foroni, Ilaria & Avellone, Alessandro & Panchuk, Anastasiia, 2015. "Sudden transition from equilibrium stability to chaotic dynamics in a cautious tâtonnement model," Chaos, Solitons & Fractals, Elsevier, vol. 79(C), pages 105-115.
    4. Brandon, Quentin & Ueta, Tetsushi & Fournier-Prunaret, Danièle & Kousaka, Takuji, 2009. "Numerical bifurcation analysis framework for autonomous piecewise-smooth dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 187-201.
    5. Radi, Davide & Gardini, Laura, 2015. "Entry limitations and heterogeneous tolerances in a Schelling-like segregation model," Chaos, Solitons & Fractals, Elsevier, vol. 79(C), pages 130-144.
    6. 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.
    7. Brianzoni, Serena & Campisi, Giovanni, 2020. "Dynamical analysis of a financial market with fundamentalists, chartists, and imitators," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).

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