Seminar Series in Quantitative Life Sciences and Medicine

“When slow meets global: geometric insight from numerics”

Hinke Osinga, University of Auckland (joint work with José Mujica (Valparaíso) and Bernd Krauskopf (Auckland))

Tuesday, September 11, 12-1pm

McIntyre Building, Room 1027

Abstract: Global manifolds are the backbone of a dynamical system and key to the characterisation of its behaviour. They arise in the classical sense of invariant manifolds associated with saddle-type equilibria or periodic orbits and also in the form of finite-time invariant manifolds in systems that evolve on multiple time scales. The latter are known as slow manifolds, because the flow along such manifolds is very slow compared with the rest of the dynamics. Slow manifolds are known to organize the number of small oscillations of so-called mixed-mode oscillations (MMOs). Their interactions with global invariant manifolds produce complicated dynamics about which only little is known from a few examples in the literature. Both global and slow manifolds need to be computed numerically. We developed accurate numerical methods based on two-point boundary value problem continuation, which have the major advantage that they remain well posed in parameter regimes where the time-scale separation varies. These techniques are particularly useful when studying changes in the global system dynamics, such as MMOs, global re-injection mechanisms, transient bursting, and phase sensitivity. This talk will focus on a transition through a quadratic tangency between the global unstable manifold of a saddle-focus equilibrium and a repelling slow manifold in a slow-fast system at the onset of MMO behaviour; more precisely, just as the equilibrium undergoes a supercritical singular Hopf bifurcation. 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 discuss recurrent dynamics in the form of MMOs, which can be continued in parameters to Shilnikov homoclinic bifurcations.”