Prof. Dipl.Ing. Dr. Peter Szmolyan (TU Wien): Multiple Time Scale Dynamical Systems

Mittwoch, 20.04.2016 16:30 im Raum M5
Mathematik und Informatik

In many ordinary differential equations models multiple time scale dynamics occurs due to the presence of variables and parameters of very different orders of magnitudes. Such problems are known collectively as singular perturbation problems or slow-fast systems. Due to an approximate splitting in slow and fast subsystems the dynamics can often be understood well enough to explain and analyse the occurrence of fairly complicated dynamical phenomena, e.g. excitability, relaxation oscillations, bursting, mixed mode oscillations, synchronization, and chaotic behaviour. In this talk a dynamical systems approach to slow-fast systems known as "Geometric Singular Perturbation Theory" (GSPT) is presented. Central objects in GSPT are "slow manifolds", which are invariant manifolds capturing the slow dynamics. Situations with a clear "global" separation into fast and slow variables correpond to singularly perturbed ordinary differential equations in standard form. For systems in standard form GSPT has been developed in great detail during the last 20 years and has found many applications in e.g. biology, chemistry, uid dynamics, and mechanics. For multi-scale problems depending on several parameters it can already be a nontrivial task to identify meaningful scalings. Typically these scalings and the corresponding asymptotic regimes are valid only in certain regions in phase-space or parameter-space. The governing equations are not globally in the standard form of slow-fast systems. An important issue is how to match these asymptotic regimes to understand the global dynamics. In the context of selected examples it will be shown that geometric methods based on GSPT and the more recently developed "blow-up method" provide a powerful approach to problems of this type.

Angelegt am Donnerstag, 07.04.2016 08:26 von cgiet_01
Geändert am Dienstag, 19.04.2016 07:42 von cgiet_01
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