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

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

doi:10.3934/dcds.2011.29.1309

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October 2011, 29(4): 1309-1344. doi: 10.3934/dcds.2011.29.1309

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

Pablo Aguirre ^1, , Eusebius J. Doedel ^2, , Bernd Krauskopf ^3, and Hinke M. Osinga ^3,

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

Received December 2009 Revised July 2010 Published December 2010

Abstract
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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, boundary value problem., invariant manifolds.

Mathematics Subject Classification: Primary: 34C45, 34C37; Secondary: 65L1.

Citation: Pablo Aguirre, Eusebius J. Doedel, Bernd Krauskopf, Hinke M. Osinga. Investigating the consequences of global bifurcations for two-dimensional invariant manifolds of vector fields. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1309-1344. doi: 10.3934/dcds.2011.29.1309

References:

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[23]	D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii and G. Huyet, Excitability in a quantum dot semiconductor laser with optical injection, Phys. Rev. Lett., 98 (2007), 153903. Google Scholar
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[26]	M. E. Henderson, Multiple parameter continuation: Computing implicitly defined $k$-manifolds, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12 (2002), 451-476. Google Scholar
[27]	M. E. Henderson, Computing invariant manifolds by integrating fat trajectories, SIAM J. Appl. Dyn. Sys., 4 (2005), 832-882. Google Scholar
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[29]	A. J. Homburg and B. Krauskopf, Resonant homoclinic flip bifurcations, J. Dynam. Diff. Eqs, 12 (2000), 807-850. Google Scholar
[30]	A. J. Homburg and B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields,, in B. Fiedler (Ed.), (). Google Scholar
[31]	J. L. Kaplan and J. A. Yorke, Preturbulence: A regime observed in a fluid flow model of Lorenz, Commun. Math. Phys., 67 (1979), 93-108. Google Scholar
[32]	B. Krauskopf and H. M. Osinga, Growing 1D and quasi-2D unstable manifolds of maps, J. Comput. Phys., 146 (1998), 406-419. Google Scholar
[33]	B. Krauskopf and H. M. Osinga, Two-dimensional global manifolds of vector fields, Chaos, 9 (1999), 768-774. Google Scholar
[34]	B. Krauskopf and H. M. Osinga, Computing geodesic level sets on global (un)stable manifolds of vector fields, SIAM J. Appl. Dyn. Sys., 2 (2003), 546-569. Google Scholar
[35]	B. Krauskopf and H. M. Osinga, Computing invariant manifolds via the continuation of orbit segments, in "Numerical Continuation Methods for Dynamical Systems," (eds. B. Krauskopf, H. M. Osinga and J. Galán-Vioque), Underst. Complex Syst., Springer-Verlag, New York, (2007), 117-154. Google Scholar
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