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Structurally unstable regular dynamics in 1D piecewise smooth maps, and circle maps

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  • Gardini, Laura
  • Tramontana, Fabio

Abstract

In this work we consider a simple system of piecewise linear discontinuous 1D map with two discontinuity points: X′=aXif ∣X∣z, where a and b can take any real value,and may have several applications. We show that its dynamic behaviors are those of a linear rotation:either periodic or quasiperiodic, and always structurally unstable. A generalization to piecewise monotone functions X′=F(X) if ∣X∣z is also given, proving the conditions leading to a homeomorphism of the circle.

Suggested Citation

  • Gardini, Laura & Tramontana, Fabio, 2012. "Structurally unstable regular dynamics in 1D piecewise smooth maps, and circle maps," Chaos, Solitons & Fractals, Elsevier, vol. 45(11), pages 1328-1342.
  • Handle: RePEc:eee:chsofr:v:45:y:2012:i:11:p:1328-1342
    DOI: 10.1016/j.chaos.2012.07.007
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    References listed on IDEAS

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    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. 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.
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    Cited by:

    1. 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.

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