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Border collision bifurcations in one-dimensional linear-hyperbolic maps


  • Gardini, Laura
  • Tramontana, Fabio
  • Sushko, Iryna


In this paper we consider a continuous one-dimensional map, which is linear on one side of a generic kink point and hyperbolic on the other side. This kind of map is widely used in the applied context. Due to the simple expression of the two functions involved, in particular cases it is possible to determine analytically the border collision bifurcation curves that characterize the dynamic behaviors of the model. In the more general model we show that the steps to be performed are the same, although the analytical expressions are not given in explicit form.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:matcom:v:81:y:2010:i:4:p:899-914
    DOI: 10.1016/j.matcom.2010.10.001

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    References listed on IDEAS

    1. Day, Richard H. & Pianigiani, Giulio, 1991. "Statistical dynamics and economics," Journal of Economic Behavior & Organization, Elsevier, vol. 16(1-2), pages 37-83, July.
    2. Zhusubaliyev, Zhanybai T. & Soukhoterin, Evgeniy & Mosekilde, Erik, 2007. "Quasiperiodicity and torus breakdown in a power electronic dc/dc converter," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 73(6), pages 364-377.
    3. Day, Richard H. & Pianigiani, Giulio, 1991. "Statistical Dynamics and Economics," Working Paper Series 293, Research Institute of Industrial Economics.
    4. Day, Richard H, 1982. "Irregular Growth Cycles," American Economic Review, American Economic Association, vol. 72(3), pages 406-414, June.
    5. 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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    Cited by:

    1. 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.
    2. 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.
    3. Gavagsaz-Ghoachani, R. & Phattanasak, M. & Martin, J.-P. & Pierfederici, S. & Davat, B., 2013. "Predicting the onset of bifurcation and stability study of a hybrid current controller for a boost converter," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 91(C), pages 262-273.
    4. Tramontana, Fabio & Westerhoff, Frank & Gardini, Laura, 2015. "A simple financial market model with chartists and fundamentalists: Market entry levels and discontinuities," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 108(C), pages 16-40.


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