Journal of Computational Dynamics (JCD)

Continuation and collapse of homoclinic tangles

Pages: 71 - 109, Volume 1, Issue 1, June 2014      doi:10.3934/jcd.2014.1.71

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Wolf-Jürgen Beyn - Department of Mathematics, Bielefeld University, P.O. Box 100131, 33501 Bielefeld, Germany (email)
Thorsten Hüls - Department of Mathematics, Bielefeld University, POB 100131, 33501 Bielefeld, Germany (email)

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.

Keywords:  Homoclinic tangency, symbolic dynamics, numerical continuation, bifurcation of homoclinic orbits.
Mathematics Subject Classification:  37N30, 65P20, 65P30.

Received: February 2011;      Revised: June 2012;      Available Online: April 2014.