Applied Maths Seminar: Szmolyan -- Hidden slow manifolds in multiple time-scale systems

Peter Szmolyan (TU Wien) is visiting us this month and will give a talk this Wednesday
25th March at 1pm in Carslaw 451.  Afterwards we will go to lunch somewhere nearby.  

Title: Hidden slow manifolds in multiple time-scale systems 

Abstract: Many biological and physical systems evolve on several disparate timescales,
i.e.  the observed dynamics have distinct temporal features that can be attributed to
processes evolving on different timescales.  In mathematical terms, such
multi-time-scale models can be considered as singular perturbation problems.  

The powerful tools of Geometric Singular Perturbation Theory (GSPT) have been widely
used to analyse complicated dynamical phenomena in many areas ranging from mathematical
physiology & neuroscience and chemical reaction systems to mechanics, fluid dynamics and
nonlinear wave problems.  

During the last years, the toolbox of GSPT has been adapted and extended to situations
where the slow-fast structures and the resulting applicability of GSPT are somewhat
hidden.  Problems of this type include singularly perturbed systems 

- in non-standard form, 

- with different multi-time-scale structure in distinct regions of phase space, 

- depending singularly on more than one parameter, 

- limiting on non-smooth systems as a parameter tends to zero.  

In situations where several distinct scalings must be used, the blow-up method has been
established as a powerful tool for matching of the different scaling regimes.  Roughly
speaking, blow-up replaces a singular point by a higher-dimensional “sphere of
directions.” This resolves the local geometry and helps track how trajectories
transition between different scaling regimes.  In this talk I will survey some of these
developments mostly in the context of specific models.