Tangencies Between Global Invariant Manifolds and Slow Manifolds Near a Singular Hopf Bifurcation

Abstract

Invariant manifolds of equilibria and periodic orbits are key objects that organize the behavior of a dynamical system both locally and globally. If multiple time scales are present in the dynamical system, there also exist so-called slow manifolds, that is, manifolds along which the flow is very slow compared with the rest of the dynamics. In particular, slow manifolds are known to organize the number of small oscillations of what are known as mixed-mode oscillations (MMOs). Slow manifolds are locally invariant objects that may interact with invariant manifolds, which are globally invariant objects; such interactions produce complicated dynamics about which only little is known from a few examples in the literature. We study the transition through a quadratic tangency between the unstable manifold of a saddle-focus equilibrium and a repelling slow manifold in a system where the corresponding equilibrium point undergoes a supercritical singular Hopf bifurcation. We compute the manifolds as families of orbit segments with a two-point boundary value problem setup and track their intersections, referred to as connecting canard orbits, as a parameter is varied. We describe the local and global properties of the manifolds, as well as the role of the interaction as an organizer of large-amplitude oscillations in the dynamics. We find and describe recurrent dynamics in the form of MMOs, which can be continued in parameters to Shilnikov homoclinic bifurcations. We detect and identify two such homoclinic orbits and describe their interactions with the MMOs.