Bifurcations of Two-Dimensional Global Invariant Manifolds near a Noncentral Saddle-Node Homoclinic Orbit

The aim of this paper is to investigate the role of the two-dimensional global invariant manifolds near a codimension-two noncentral saddle-node homoclinic point in a three-dimensional vector field. The main question is to determine how the arrangement of global two-dimensional manifolds changes through the unfolding and how this affects the topological organization of basins of attraction. To this end, we compute the respective global invariant manifolds---rendered as surfaces in the three-dimensional phase space---and their intersection curves with a suitable sphere as families of orbit segments with a two-point boundary value problem setup. As a specific example to work on, we consider a laser model with optical injection which undergoes this codimension-two bifurcation. We first investigate the transition through each codimension-one bifurcation that occurs near a noncentral saddle-node homoclinic point; in particular, we show how the basins of attraction of the bifurcating periodic orbit Γ and of an attractor q are created in each case and how their basin boundaries are formed by the stable manifold Wˢ(p) of a saddle-focus p and by the two-dimensional strong stable manifold Wˢˢ(q) of q. In particular, in the case of a local saddle-node bifurcation of equilibria, as well as in a saddle-node homoclinic bifurcation, we explain how Wˢ(p) and Wˢˢ(q) collide with one another and disappear, in that it is, effectively, a saddle-node bifurcation of global two-dimensional invariant manifolds. We then study the global invariant manifolds in the transition at the codimension-two point. More specifically, we present two-parameter bifurcation diagrams of the codimension-two singularity with representative images, in phase space and on the sphere, of Wˢ(p) and Wˢˢ(q) and the relevant basins of attraction. In particular, we identify conditions for multipulse behavior in the laser model depending on the global manifolds Wˢ(p) and Wˢˢ(q) in open regions of parameter space and at the bifurcations involved.