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

Abstract

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 $\Gamma$ and of an attractor $\mathbf{q}$ are created in each case and how their basin boundaries are formed by the stable manifold $W^s(\mathbf{p})$ of a saddle-focus $\mathbf{p}$ and by the two-dimensional strong stable manifold $W^{ss}(\mathbf{q})$ of $\mathbf{q}$. 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 $W^s(\mathbf{p})$ and $W^{ss}(\mathbf{q})$ 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 $W^s(\mathbf{p})$ and $W^{ss}(\mathbf{q})$ and the relevant basins of attraction. In particular, we identify conditions for multipulse behavior in the laser model depending on the global manifolds $W^s(\mathbf{p})$ and $W^{ss}(\mathbf{q})$ in open regions of parameter space and at the bifurcations involved.