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Abstract
We consider a homoclinic bifurcation of a vector field in [\R^3] , where a onedimensional unstable manifold of an equilibrium is contained in the twodimensional stable manifold of this same equilibrium. How such onedimensional 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 twodimensional 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 twodimensional manifolds, and their onedimensional intersection curves with a suitable crosssection, 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 twodimensional 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.
Translated title of the contribution  Investigating the consequences of global bifurcations for twodimensional invariant manifolds of vector fields 

Original language  English 
Pages (fromto)  1309  1344 
Number of pages  35 
Journal  Discrete and Continuous Dynamical Systems (DCDSA)  Series A 
Volume  29 Issue 4 
DOIs  
Publication status  Published  Dec 2010 
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Projects
 1 Finished

GLOBAL INVARIANT MANIFOLDS: APPLICATIONS, CRITICAL BOUNDERIES AND GLOBAL BIFURCATIONS (ADVANCED RESEARCH FELLOWSHIP)
Osinga, H. M.
1/10/05 → 1/06/11
Project: Research