Global Invariant Manifolds and their Interactions in the Neighborhood of a Homoclinic Flip Bifurcation

Abstract

This thesis deals with bifurcations of vector fields, which occur in models of choice in numerous applications. More specifically, we are concerned with a type of global bifurcation called homoclinic flip bifurcation, which is known to generate complicated dynamics. When a real saddle equilibrium in a three-dimensional vector field undergoes a codimension-one homoclinic bifurcation, the associated two-dimensional invariant manifold of the equilibrium closes on itself in an orientable or nonorientable way. The main focus of this thesis is to study how the global invariant manifolds of different saddle periodic orbits and equilibria reorganize phase space for a vector field close to a codimension-two homoclinic flip bifurcation. This is the point of transition between having the orientable and the nonorientable case. Such a codimension-two homoclinic flip bifurcation point unfolds generically in three different cases, denoted A, B and C; case A has been considered previously. In this thesis, we focus on cases B and C, which are organizing centers for the creation and disappearance of saddle periodic orbits (a new feature compared to case A). To explain how the global manifolds organize phase space, we consider Sandstede’s threedimensional vector field model which features these bifurcations. We compute global invariant manifolds and their intersection sets with a suitable sphere by means of continuation of suitable two-point boundary problems. In this way, we are able to understand their variations in geometry as different codimension-one bifurcations are crossed and their roles as separatrices of basins of attracting periodic orbits. We present the unfoldings of both cases B and C in unprecedented detail. In particular, we find heteroclinic orbits between saddle periodic orbits and equilibria, which give rise to regions of infinitely many heteroclinic orbits for both cases B and C. In particular, we identify and characterize conjectured results about chaotic and sensitive dynamics for case C; furthermore, we also discover a plethora of additional bifurcation phenomena for both cases. Overall, we present a geometric picture of the many different bifurcations involved. Apart from being of interest for completing the theory of homoclinic flip bifurcations, our results and associated numerical methods are also relevant for models that describe physical phenomena, like the Hindmarsh–Rose model and the Van der Pol–Duffing model, where homoclinic flip bifurcations have been identified as important ingredients to explain observed behaviors.