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Bifurcations in Smooth and Piecewise Smooth Noninvertible Maps

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Difference Equations, Discrete Dynamical Systems and Applications (ICDEA 2017)

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Abstract

In the present chapter we recall some basic concepts and results associated with the local and global bifurcations of attractors and their basins in smooth and nonsmooth noninvertible maps, continuous and discontinuous. Such maps appear to be important both from theoretical and applied points of view. Using numerous examples we show that noninvertibility and nonsmoothness, as well as discontinuity of the considered maps lead to peculiar bifurcation phenomena which cannot be observed in smooth invertible maps.

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Notes

  1. 1.

    Recall that a critical set LC of a continuous noninvertible map F is defined as the locus of points having at least two coincident rank-1 preimages located on the set \(LC_{-1}\) (which belongs to the set of vanishing determinant of a smooth map, or to the set of nonsmoothness of a piecewise smooth map), \(LC=F(LC_{-1})\). The simplest example is a local extremum of a 1D map which is a critical point of this map.

  2. 2.

    A fixed point or cycle is called expanding if all its eigenvalues of are larger than 1 in modulus.

  3. 3.

    For the proper first return map between the regions corresponding to the cycles with rotation numbers \(m_{1}/n_{1}\) and \(m_{2}/n_{2}\) which are Farey neighbors (i.e., \(\left| m_{1}n_{2}-m_{2}n_{1}\right| =1\)) there exists a region of cycles with rotation number \((m_{1}+m_{2})/(n_{1}+n_{2}).\)

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Authors and Affiliations

  1. Department of Economics, Society, Politics, University of Urbino, Urbino, Italy

    Laura Gardini

  2. Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    Iryna Sushko

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  1. Laura Gardini
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Correspondence to Laura Gardini .

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  1. Department of Mathematics, Trinity University, San Antonio, TX, USA

    Saber Elaydi

  2. Institut für Mathematik, Alpen-Adria Universität Klagenfurt, Klagenfurt, Austria

    Christian Pötzsche

  3. Department of Mathematics, West University of TimiÅŸoara, TimiÅŸoara, Romania

    Adina Luminiţa Sasu

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Gardini, L., Sushko, I. (2019). Bifurcations in Smooth and Piecewise Smooth Noninvertible Maps. In: Elaydi, S., Pötzsche, C., Sasu, A. (eds) Difference Equations, Discrete Dynamical Systems and Applications. ICDEA 2017. Springer Proceedings in Mathematics & Statistics, vol 287. Springer, Cham. https://doi.org/10.1007/978-3-030-20016-9_4

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