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Contact bifurcations related to critical sets and focal points in iterated maps of the plane. (English) Zbl 1242.39024
Liz, Eduardo (ed.) et al., Proceedings of the workshop on future directions in difference equations, Vigo, Spain, June 13–17, 2011. Vigo: Servizo de Publicacións da Universidade de Vigo (ISBN 978-84-8158-541-4/pbk). Colección Congresos 69, 15-50 (2011).
Summary: In this survey article we briefly describe some properties of difference equations obtained by the iterated applications of two-dimensional maps of the plane and we try to characterize the qualitative changes (or bifurcations) of the asymptotic behavior of the solutions, as some parameters are varied, in terms of contacts between particular curves and invariant sets which characterize the global properties of the iterated maps. In particular we consider the role of critical curves in noninvertible maps of the plane and the contacts involving focal points and prefocal sets in maps of the plane characterized by the presence (in the map or in some inverse map) of a denominator that vanishes along a curve of the plane. The effects of the global bifurcations given by contacts involving such singular curves on the attractors and their basins are described through some examples.
MSC:
39A28Bifurcation theory (difference equations)
39-02Research monographs (functional equations)
39A10Additive difference equations
37G35Attractors and their bifurcations
39A12Discrete version of topics in analysis