Prof. Dipl.Ing. Dr. Peter Szmolyan (TU Wien): Multiple Time Scale Dynamical Systems
Wednesday, 20.04.2016 16:30 im Raum M5
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
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.