Global Invariant Manifolds Near Homoclinic Orbits to a Real Saddle: (Non)Orientability and Flip Bifurcation

Homoclinic bifurcations are important phenomena that cause global re-arrangements of the dynamics in phase space, including changes to basins of attractions and the generation of chaotic dynamics. We consider here a homoclinic (or connecting) orbit that converges in both forward and backward time to a saddle equilibrium of a three-dimensional vector field. We assume that the saddle is such that the eigenvalues of its Jacobian are real. If such a homoclinic orbit is broken by varying a suitable parameter then, generically, a single periodic orbit Γ bifurcates. We consider the case that the saddle quantity of the equilibrium is negative so that Γ is an attractor (rather than of saddle type). At the moment of bifurcation the two-dimensional stable manifold of the saddle, when followed along the homoclinic orbit, may form either an orientable or nonorientable surface, and one speaks of an orientable or a nonorientable homoclinic bifurcation. A change of orientability occurs at two kinds of codimension-two homoclinic bifurcations, namely, an inclination flip and an orbit flip. The stable manifold of the saddle point is neither orientable nor nonorientable at either of these bifurcations. In this paper we study how the stable manifold of the saddle organizes the phase space globally near these homoclinic bifurcations. To this end, we consider a model vector field due to Sandstede, in which the origin 0 is a saddle point that undergoes the respective homoclinic bifurcations for certain choices of the parameters. We compute its global stable manifold W (0) via the continuation of suitable orbit segments to determine how it changes through the bifurcation in question. More specifically, we render W (0) as a two-dimensional surface in the three-dimensional phase space, and also consider its intersection set with a suitable sphere. We first investigate the transition through the orientable and nonorientable codimension-one homoclinic bifurcations (with negative saddle quantity); in particular, we show how the basin of attraction of the bifurcating periodic orbit Γ is created in each case. We then study the global invariant manifold W (0) near the transition between these two cases as given by an inclination flip and an orbit flip bifurcation. More specifically, we present two-parameter bifurcation diagrams of the two flip bifurcations with representative images, in phase space and on the sphere, of W (0) in relation to other relevant invariant objects. In this way, we identify the topological properties of W (0) in open regions of parameter space and at the bifurcations involved. ∗Group of Analysis and Mathematical Modeling Valparáıso AM2V, Departamento de Matemática, Universidad Técnica Federico Santa Maŕıa, Casilla 110-V, Valparáıso, Chile ([email protected]) †Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand ([email protected]; [email protected]) ‡Partially funded by FONDECYT Postdoctoral Grant No. 3130497 and Proyecto Basal CMM Universidad de Chile. This work was carried out at the Department of Engineering Mathematics, University of Bristol with support from a CONICYT grant and an ORS award; this author also acknowledges the hospitality of The University of Auckland during two longer research visits.