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Homoclinic bifurcations are important phenomena that cause global rearrangements 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 s (0) via the continuation of suitable orbit segments to determine how it changes through the bifurcation in question. More specifically, we render W s (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 s (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 s (0) in relation to other relevant invariant objects. In this way, we identify the topological properties of W s (0) in open regions of parameter space and at the bifurcations involved.

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We consider a three-dimensional vector field with a Shilnikov homoclinic orbit that converges to a saddle-focus equilibrium in both forward and backward time. The one-parameter unfolding of this global bifurcation depends on the sign of the saddle quantity. When it is negative, breaking the homoclinic orbit produces a single stable periodic orbit; this is known as the simple Shilnikov bifurcation. However, when the saddle quantity is positive, the mere existence of a Shilnikov homoclinic orbit induces complicated dynamics, and one speaks of the chaotic Shilnikov bifurcation; in particular, one finds suspended horseshoes and countably many periodic orbits of saddle type. These well-known and celebrated results on the Shilnikov homoclinic bifurcation have been obtained by the classical approach of reducing a Poincaré return map to a one-dimensional map.In this paper, we study the implications of the transition through a Shilnikov bifurcation for the overall organization of the three-dimensional phase space of the vector field. To this end, we focus on the role of the two-dimensional global stable manifold of the equilibrium, as well as those of bifurcating saddle periodic orbits. We compute the respective two-dimensional global manifolds, and their intersection curves with a suitable sphere, as families of orbit segments with a two-point boundary-value-problem setup. This allows us to determine how the arrangement of global manifolds changes through the bifurcation and how this influences the topological organization of phase space. For the simple Shilnikov bifurcation, we show how the stable manifold of the saddle focus forms the basin boundary of the bifurcating stable periodic orbit. For the chaotic Shilnikov bifurcation, we find that the stable manifold of the equilibrium is an accessible set of the stable manifold of a chaotic saddle that contains countably many periodic orbits of saddle type. In intersection with a suitably chosen sphere we find that this stable manifold is an indecomposable continuum consisiting of infinitely many closed curves that are locally a Cantor bundle of arcs.