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

Laura Gardini (), Roya Makrooni () and Iryna Sushko ()
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Roya Makrooni: Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran
Iryna Sushko: Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

No 201603, Gecomplexity Discussion Paper Series from Action IS1104 "The EU in the new complex geography of economic systems: models, tools and policy evaluation"

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.

Keywords: Piecewise smooth maps; Border collision bifurcations; Cascade of smooth and nonsmooth bifurcations; Skew tent map as a normal form; Unbounded chaotic sets (search for similar items in EconPapers)
Pages: 30 pages
Date: 2016-03, Revised 2016-03
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http://www.gecomplexity-cost.eu/repec/cst/wpaper/geco_dp_3_2016.pdf First version, 2016 (application/pdf)

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Persistent link: https://EconPapers.repec.org/RePEc:cst:wpaper:201603

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