Continuation and collapse of homoclinic tangles

Wolf-Jürgen Beyn; Thorsten Hüls

doi:10.3934/jcd.2014.1.71

Article Contents

2014, Volume 1, Issue 1: 71-109. Doi: 10.3934/jcd.2014.1.71

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Continuation and collapse of homoclinic tangles

Wolf-Jürgen Beyn^1, and
Thorsten Hüls^2,

1.
Department of Mathematics, Bielefeld University, P.O. Box 100131, 33501 Bielefeld
2.
Department of Mathematics, Bielefeld University, POB 100131, 33501 Bielefeld

Received: January 31, 2011

Revised: May 31, 2012

Published: December 2013

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Abstract

By a classical theorem transversal homoclinic points of maps lead to shift dynamics on a maximal invariant set, also referred to as a homoclinic tangle. In this paper we study the fate of homoclinic tangles in parameterized systems from the viewpoint of numerical continuation and bifurcation theory. The new bifurcation result shows that the maximal invariant set near a homoclinic tangency, where two homoclinic tangles collide, can be characterized by a system of bifurcation equations that is indexed by a symbolic sequence. These bifurcation equations consist of a finite or infinite set of hilltop normal forms known from singularity theory. For the Hénon family we determine numerically the connected components of branches with multi-humped homoclinic orbits that pass through several tangencies. The homoclinic network found by numerical continuation is explained by combining our bifurcation result with graph-theoretical arguments.

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Mathematics Subject Classification: 37N30, 65P20, 65P30.

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