TY - UNPB

T1 - Investigating the consequences of global bifurcations for two-dimensional invariant manifolds of vector fields

AU - Aguirre Olea, PL

AU - Doedel, EJ

AU - Krauskopf, B

AU - Osinga, HM

N1 - Sponsorship: The research of P.A. was supported by a CONICYT and an ORS grant, that of E.J.D. by an NSERC (Canada) Discovery Grant, and that of H.M.O. by an EPSRC Advanced Research Fellowship grant

PY - 2009/11

Y1 - 2009/11

N2 - 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 two-dimensional stable manifold changes in the process. 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.

AB - 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 two-dimensional stable manifold changes in the process. 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.

M3 - Working paper and Preprints

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

ER -