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Border collision bifurcation curves and their classification in a family of 1D discontinuous maps


  • Gardini, Laura
  • Tramontana, Fabio


In this paper we consider a one-dimensional piecewise linear discontinuous map in canonical form, which may be used in several physical and engineering applications as well as to model some simple financial markets. We classify three different kinds of possible dynamic behaviors associated with the stable cycles. One regime (i) is the same existing in the continuous case and it is characterized by periodicity regions following the period increment by 1 rule. The second one (ii) is the regime characterized by periodicity regions of period increment higher than 1 (we shall see examples with 2 and 3), and by bistability. The third one (iii) is characterized by infinitely many periodicity regions of stable cycles, which follow the period adding structure, and multistability cannot exist. The analytical equations of the border collision bifurcation curves bounding the regions of existence of stable cycles are determined by using a new approach.

Suggested Citation

  • Gardini, Laura & Tramontana, Fabio, 2011. "Border collision bifurcation curves and their classification in a family of 1D discontinuous maps," Chaos, Solitons & Fractals, Elsevier, vol. 44(4), pages 248-259.
  • Handle: RePEc:eee:chsofr:v:44:y:2011:i:4:p:248-259
    DOI: 10.1016/j.chaos.2011.02.001

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    References listed on IDEAS

    1. Tramontana, Fabio & Westerhoff, Frank & Gardini, Laura, 2010. "On the complicated price dynamics of a simple one-dimensional discontinuous financial market model with heterogeneous interacting traders," Journal of Economic Behavior & Organization, Elsevier, vol. 74(3), pages 187-205, June.
    2. Day, Richard H, 1982. "Irregular Growth Cycles," American Economic Review, American Economic Association, vol. 72(3), pages 406-414, June.
    3. Viktor Avrutin & Michael Schanz & Björn Schenke, 2011. "Coexistence of the Bandcount-Adding and Bandcount-Increment Scenarios," Discrete Dynamics in Nature and Society, Hindawi, vol. 2011, pages 1-30, March.
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    Cited by:

    1. Cerboni Baiardi, Lorenzo & Naimzada, Ahmad K. & Panchuk, Anastasiia, 2020. "Endogenous desired debt in a Minskyan business model," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    2. 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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