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Cascades of Alternating Smooth Bifurcations and Border Collision Bifurcations in a Family of Discontinuous Linear-Power Maps

Author

Listed:
  • Laura Gardini

    (DESP, University of Urbino Carlo Bo, Urbino, Italy)

  • Roya Makrooni

    (Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran)

  • Iryna Sushko

    (Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine)

Abstract

In this work we investigate dynamics of a family of one-dimensional discontinuous linear-power maps. This family in the continuous case associated with Nordmark systems has been studied by many authors. Recently, also the discontinuous case has been considered, in which the map has a vertical asymptote giving rise to new kinds of bifurcations. In the considered family, a transition from invertible to non-invertible map may lead abruptly to chaos, and the role of organizing center in the parameter space is played by a particular codimension-two bifurcation point related to this transition and to a ‡ip bifurcation. The interplay between smooth bifurcations and border collision bifurcations is described explaining the peculiar periods of attracting cycles appearing due to nonstandard cascades of bifurcations. Robust unbounded chaotic attractors characteristic for certain parameter ranges are also described. We provide proofs of some properties of the considered map, however, complete description of its rich bifurcation structure is still an open problem.

Suggested Citation

  • 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.
    • Handle: RePEc:cst:wpaper:201603
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    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. Makrooni, Roya & Abbasi, Neda & Pourbarat, Mehdi & Gardini, Laura, 2015. "Robust unbounded chaotic attractors in 1D discontinuous maps," Chaos, Solitons & Fractals, Elsevier, vol. 77(C), pages 310-318.
    4. Sushko, Iryna & Gardini, Laura & Matsuyama, Kiminori, 2014. "Superstable credit cycles and U-sequence," Chaos, Solitons & Fractals, Elsevier, vol. 59(C), pages 13-27.
    5. Gardini, Laura & Sushko, Iryna & Avrutin, Viktor & Schanz, Michael, 2011. "Critical homoclinic orbits lead to snap-back repellers," Chaos, Solitons & Fractals, Elsevier, vol. 44(6), pages 433-449.
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