Title:

The role of global invariant manifolds of vector fields at homoclinic bifurcations

We consider certain kinds of homoclinic bifurcations in threedimensional vector fields. These global bifurcations are characterized by the existence of a homo clinic orbit that converges to a saddle equilibrium in both forward and backward time. If the equilibrium has a complex pair of (stable) eigenvalues, it is a saddlefocus, and one speaks of a Shilnikov homoclinic orbit. In this case, the homoclinic orbit converges towards the equilibrium in a spiralling fashion. On the other hand, if the saddle equilibrium has two real (stable) eigenvalues, then the homoclinic orbit converges generically to the saddle along the direction given by the weak stable eigenvector. The possible unfoldings of a codimensionone homoclinic bifurcation depend on the sign of the saddle quantity: when it is negative, breaking the homoclinic orbit results in a single stable periodic orbit from a saddlefocus homoclinic orbit; one speaks of a simple Shilnikov bifurcation. However, when the saddle quantity is positive, then the mere existence of a Shilnikov homoclinic orbit induces complicated dynamics, and one speaks of a chaotic Shilnikov bifurcation. For a homoclinic orbit to a real saddle, on the other hand, always a single periodic orbit bifurcates, which is attracting when the saddle quantity is negative and of saddle type when it is positive. In this thesis we show how the global threedimensional phase space is organized near certain homoclinic bifurcations by the twodimensional global stable manifolds of equilibria and periodic orbits. To this end, we consider a model of a laser with optical injection that contains Shilnikov homoclinic orbits and a model by Sandstede that features different kinds of homoclinic bifurca tions to a saddle. We find that, in the simple Shilnikov case, the stable manifold ofthe saddlefocus forms the basin boundary of the bifurcating stable periodic orbit. On the other hand, in the chaotic case, the stable manifold of the equilibrium is the accessible set of a chaotic saddle that contains countably many periodic orbits of saddle type. In the case of a homoclinic bifurcation to a saddle, the stable manifold of the saddle is either an orientable or nonorientable twodimensional surface. A change of orientability occurs at two kinds of codimensiontwo homoclinic bifurcations, called inclination flip and orbit flip bifurcations. At either of these flip bifurcation points, the stable manifold is neither orientable nor nonorientable, but just at the transition between both states. We show how this transition occurs for the case of negative saddle quantity, and how the basin of attraction of the stable periodic orbit is organized in different ways by the stable manifold of the saddle depending on the (non)orientability of the bifurcation. Finally, we show how the stable manifold rearranges both itself and the overall dynamics in phase space near the codimensiontwo transition from a saddle to saddlefocus homoclinic bifurcation that occurs at a socalled Belyakov point.
