Bifurcation of Connecting Orbits with One Nonhyperbolic Fixed Point for Maps

In this paper we consider the bifurcation of transversal heteroclinic orbits in discrete time dynamical systems. We assume that a nonhyperbolic transversal heteroclinic orbit exists at some critical parameter value. This situation appears, for example, when one end point undergoes a fold or flip bif...

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Bibliographic Details
Published inSIAM journal on applied dynamical systems Vol. 4; no. 4; pp. 985 - 1007
Main Author Huls, Thorsten
Format Journal Article
LanguageEnglish
Published Philadelphia Society for Industrial and Applied Mathematics 01.01.2005
Subjects
Approximation
Dynamical systems
Graph representations
Neighborhoods
Numerical analysis
Orbits
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Summary:In this paper we consider the bifurcation of transversal heteroclinic orbits in discrete time dynamical systems. We assume that a nonhyperbolic transversal heteroclinic orbit exists at some critical parameter value. This situation appears, for example, when one end point undergoes a fold or flip bifurcation. In these two cases the bifurcation analysis of the orbit is performed in detail. In particular, we prove, using implicit function techniques, that the orbit can be continued beyond the bifurcation point. Finally, we show numerical computations for the fold and for the flip bifurcations.
ISSN:1536-0040
1536-0040
DOI:10.1137/040614670