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

P. Aguirre, Doedel E. J., B Krauskopf, HM Osinga

Research output: Contribution to journalArticle (Academic Journal)peer-review

22 Citations (Scopus)

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.
Translated title of the contributionInvestigating the consequences of global bifurcations for two-dimensional invariant manifolds of vector fields
Original languageEnglish
Pages (from-to)1309 - 1344
Number of pages35
JournalDiscrete and Continuous Dynamical Systems (DCDS-A) - Series A
Volume29 Issue 4
DOIs
Publication statusPublished - Dec 2010

Fingerprint Dive into the research topics of 'Investigating the consequences of global bifurcations for two-dimensional invariant manifolds of vector fields'. Together they form a unique fingerprint.

Cite this