Tuesday 22 Jun 2021: Interacting manifolds shaping the dynamics: Slow-fast dynamical systems and more
Jose Mujica - VU Amsterdam
https://Universityofexeter.zoom.us/j/98070370979?pwd=SmNBQk43NlBXQk1ZV1BXUEozai9xQT09 Meeting ID: 980 7037 0979 Password: 756520 13:30-14:30
Invariant objects together with their invariant manifolds provide the skeleton of a dynamical system. They correspond to boundaries between regions on phase space with qualitatively different behaviour. Getting an understanding of invariant manifolds allow us to get a good insight into the dynamics of a system. For instance, the interactions between invariant manifolds of different type give rise to changes on the configuration of a system by means of global bifurcations; well know examples are homoclinic and heteroclinic bifurcations that arise in several applications.
In this talk we will discuss the dynamics of a family of slow-fast dynamical systems with one fast and two slow variables. The focus is initially on a (local) bifurcation that occurs in the slow dynamics known as a folded saddle-node of type II. This scenario provides a crash between classical dynamical systems and slow-fast systems, in the sense that there is an interaction of a global invariant manifold with a slow manifold. This has consequences for the local and global dynamics of the system. In particular we discuss the organisation of recurrent dynamics in the form of oscillatory patterns known as mixed-mode oscillations, and the homoclinic scenarios nearby. The way we approach this is via the numerical approximation of these manifolds in a boundary-value-problem setup with the software AUTO, and track the manifolds when system parameters are varied. We will end with a brief discussion of how some of these ideas can be implemented in the context of a network of coupled oscillators close to heteroclinic cycles with localised frequency synchrony.
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