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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

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

Pages: 1309 - 1344, Volume 29, Issue 4, April 2011      doi:10.3934/dcds.2011.29.1309

 
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Pablo Aguirre - Bristol Centre for Applied Nonlinear Mathematics, Department of Engineering Mathematics, University of Bristol, Bristol BS8 1TR, United Kingdom (email)
Eusebius J. Doedel - Department of Computer Science, Concordia University, 1455 Boulevard de Maisonneuve O., Montréal, Québec H3G 1M8, Canada (email)
Bernd Krauskopf - Bristol Center for Applied Nonlinear Mathematics, Department of Engineering Mathematics, University of Bristol, Queen's Building, Bristol BS8 1TR, United Kingdom (email)
Hinke M. Osinga - Bristol Center for Applied Nonlinear Mathematics, Department of Engineering Mathematics, University of Bristol, Queen's Building, Bristol BS8 1TR, United Kingdom (email)

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.

Keywords:  Homoclinic bifurcation, invariant manifolds, boundary value problem.
Mathematics Subject Classification:  Primary: 34C45, 34C37; Secondary: 65L10.

Received: December 2009;      Revised: July 2010;      Available Online: December 2010.

 References