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

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

Abstract

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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    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. Xue-Zhong He & Kai Li & Chuncheng Wang, 2018. "Time-varying economic dominance in financial markets: A bistable dynamics approach," Published Paper Series 2018-1, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    2. 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.
    3. Sushko, Iryna & Gardini, Laura & Matsuyama, Kiminori, 2016. "Robust chaos in a credit cycle model defined by a one-dimensional piecewise smooth map," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 299-309.
    4. Viktor Avrutin & Iryna Sushko & Fabio Tramontana, 2014. "Bifurcation Structure in a Bimodal Piecewise Linear Business Cycle Model," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    5. Spiros Bougheas & Pasquale Commendatore & Laura Gardini & Ingrid Kubin, 2025. "Dynamic Investigations of an Endogenous Business Cycle Model with Heterogeneous Agents," Computational Economics, Springer;Society for Computational Economics, vol. 65(2), pages 665-689, February.
    6. Tramontana, Fabio & Gardini, Laura & Agliari, Anna, 2011. "Endogenous cycles in discontinuous growth models," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(8), pages 1625-1639.
    7. Giovanni Campisi & Anastasiia Panchuk & Fabio Tramontana, 2024. "A discontinuous model of exchange rate dynamics with sentiment traders," Annals of Operations Research, Springer, vol. 337(3), pages 913-935, June.
    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. Brianzoni, Serena & Campisi, Giovanni, 2020. "Dynamical analysis of a financial market with fundamentalists, chartists, and imitators," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    10. Davide Radi & Laura Gardini & Viktor Avrutin, 2014. "The Role of Constraints in a Segregation Model: The Asymmetric Case," Discrete Dynamics in Nature and Society, Hindawi, vol. 2014, pages 1-17, August.
    11. Gian Italo Bischi & Lorenzo Cerboni Baiardi & Fabio Lamantia & Davide Radi, 2024. "Nonlinear dynamics and game-theoretic modeling in economics and finance," Annals of Operations Research, Springer, vol. 337(3), pages 731-737, June.
    12. Tramontana, F. & Gardini, L. & Ferri, P., 2010. "The dynamics of the NAIRU model with two switching regimes," Journal of Economic Dynamics and Control, Elsevier, vol. 34(4), pages 681-695, April.
    13. Iryna Sushko & Laura Gardini & Kiminori Matsuyama, 2014. "Chaos in a Model of Credit Cycles with Good and Bad Projects," Working Papers 1405, University of Urbino Carlo Bo, Department of Economics, Society & Politics - Scientific Committee - L. Stefanini & G. Travaglini, revised 2014.
    14. Anna Agliari & Pasquale Commendatore & Ilaria Foroni & Ingrid Kubin, 2014. "Expectations and industry location: a discrete time dynamical analysis," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 37(1), pages 3-26, April.
    15. Tramontana, Fabio & Gardini, Laura & Puu, Tönu, 2009. "Cournot duopoly when the competitors operate multiple production plants," Journal of Economic Dynamics and Control, Elsevier, vol. 33(1), pages 250-265, January.
    16. Agliari, Anna & Commendatore, Pasquale & Foroni, Ilaria & Kubin, Ingrid, 2015. "Agglomeration dynamics and first nature asymmetries," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 108(C), pages 81-98.
    17. 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.
    18. Gardini, Laura & Tramontana, Fabio & Sushko, Iryna, 2010. "Border collision bifurcations in one-dimensional linear-hyperbolic maps," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(4), pages 899-914.
    19. Dal Forno, Arianna & Merlone, Ugo, 2013. "Border-collision bifurcations in a model of Braess paradox," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 87(C), pages 1-18.
    20. Sushko, Iryna & Gardini, Laura & Matsuyama, Kiminori, 2014. "Superstable credit cycles and U-sequence," Chaos, Solitons & Fractals, Elsevier, vol. 59(C), pages 13-27.
    21. 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.
    22. Anna Agliari & Pasquale Commendatore & Ilaria Foroni & Ingrid Kubin, 2011. "Border Collision Bifurcations in a Footloose Capital Model with First Nature Firms," Computational Economics, Springer;Society for Computational Economics, vol. 38(3), pages 349-366, October.
    23. 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.
    24. 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.
    25. Gardini, Laura & Sushko, Iryna & Naimzada, Ahmad K., 2008. "Growing through chaotic intervals," Journal of Economic Theory, Elsevier, vol. 143(1), pages 541-557, November.

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