Investigating the consequences of global bifurcationsfor two-dimensional invariant manifolds of vector fields

Pablo Aguirre; Eusebius J. Doedel; Bernd Krauskopf; Hinke M. Osinga

doi:10.3934/dcds.2011.29.1309

Article Contents

2011, Volume 29, Issue 4: 1309-1344. Doi: 10.3934/dcds.2011.29.1309

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Investigating the consequences of global bifurcations for two-dimensional invariant manifolds of vector fields

1.
Bristol Centre for Applied Nonlinear Mathematics, Department of Engineering Mathematics, University of Bristol, Bristol BS8 1TR
2.
Department of Computer Science, Concordia University, 1455 Boulevard de Maisonneuve O., Montréal, Québec H3G 1M8
3.
Bristol Center for Applied Nonlinear Mathematics, Department of Engineering Mathematics, University of Bristol, Queen's Building, Bristol BS8 1TR

Received: November 30, 2009

Revised: June 30, 2010

Published: September 2011

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Abstract

We consider a homoclinic bifurcation of a vector field in $\R^3$, where a one-dimensional unstable manifold of an equilibrium is contained in the two-dimensional stable manifold of this same equilibrium. How such one-dimensional connecting orbits arise is well understood, and software packages exist to detect and follow them in parameters.
In this paper we address an issue that it is far less well understood: how does the associated two-dimensional stable manifold change geometrically during the given homoclinic bifurcation? This question can be answered with the help of advanced numerical techniques. More specifically, we compute two-dimensional manifolds, and their one-dimensional intersection curves with a suitable cross-section, via the numerical continuation of orbit segments as solutions of a boundary value problem. In this way, we are able to explain how homoclinic bifurcations may lead to quite dramatic changes of the overall dynamics. This is demonstrated with two examples. We first consider a Shilnikov bifurcation in a semiconductor laser model, and show how the associated change of the two-dimensional stable manifold results in the creation of a new basin of attraction. We then investigate how the basins of the two symmetrically related attracting equilibria change to give rise to preturbulence in the first homoclinic explosion of the Lorenz system.

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Mathematics Subject Classification: Primary: 34C45, 34C37; Secondary: 65L10.

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