91社区's Seminar Series in Quantitative Life Sciences and Medicine
Sponsored by CAMBAM, QLS, MiCM and the Ludmer Centre
Title: 鈥淲hen slow meets global: geometric insight from numerics鈥�
Speaker:听Hinke Osinga, University of Auckland (joint work with Jos茅 Mujica (Valpara铆so) and Bernd Krauskopf (Auckland))
When: Tuesday, September 11, 12-1pm
Where: 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听 听 听 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.听 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听 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听 听 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听 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.鈥�
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