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Robust unbounded chaotic attractors in 1D discontinuous maps

Author

Listed:
  • Roya Makrooni

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

  • Neda Abbasi

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

  • Mehdi Pourbarat

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

  • Laura Gardini

    (Department of Economics, Society & Politics, Università di Urbino "Carlo Bo")

Abstract

In this paper we prove the existence of full measure unbounded chaotic attractors which are persistent under parameter perturbation (also called robust). We show that this occurs in a discontinuous piecewise smooth one-dimensional map f, belonging to the family known as Nordmark's map. To prove the result we extend the properties of a full shift on a finite or infinite number of symbols to a map, here called Baker-like map with infinitely many branches, defined as a map of the interval I = [0; 1] into itself with infinitely branches due to expanding functions with range I except at most the rightmost one. The proposed example is studied by using the first return map in I, which we prove to be chaotic in I making use of the border collision bifurcations curves of basic cycles. This leads to a robust unbounded chaotic attractor, the interval (- ; 1], for the map f.

Suggested Citation

  • Roya Makrooni & Neda Abbasi & Mehdi Pourbarat & Laura Gardini, 2015. "Robust unbounded chaotic attractors in 1D discontinuous maps," Working Papers 1501, University of Urbino Carlo Bo, Department of Economics, Society & Politics - Scientific Committee - L. Stefanini & G. Travaglini, revised 2015.
    • Handle: RePEc:urb:wpaper:15_01
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    File URL: http://www.econ.uniurb.it/RePEc/urb/wpaper/WP_15_01.pdf
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    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.
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    Cited by:

    1. 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.
    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.

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    1. 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.

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