Invariant manifolds of equilibria and periodic orbits are key objects that organise the behaviour 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 organise 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 orbits segments with a twopoint 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 organiser 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.

No abstract

Canard orbits are relevant objects in slow-fast dynamical systems that organize the spiraling of orbits nearby. In three-dimensional vector fields with two slow and one fast variables, canard orbits arise from the intersection between an attracting and a repelling two-dimensional slow manifold. Special points called folded nodes generate such intersections: in a suitable transverse two-dimensional section Σ, the attracting and repelling slow manifolds are counter-rotating spirals that intersect in a finite number of points. We present an implementation of Lin's method that is able to detect all of these intersection points and, hence, all of the canard orbits arising from a folded node. With a boundary-value-problem setup we compute orbit segments on each slow manifold up to Σ, where we require that the corresponding end points in Σ lie in a one-dimensional subspace known as the Lin space Z. The Lin space Z must be transverse to the slow manifolds and it remains fixed during the detection of canard orbits as zeros of the signed distance along Z. During the computation, a tangency of Z with one of the intersection curves in Σ may arise. To overcome this, we update the Lin space at an intermediate continuation step to detect a double tangency of Z to both curves in Σ, after which the canard detection is able to continue. Our method is demonstrated with the examples of the normal form for a folded node and of the Koper model.