Summer School 2019

Andrea Braides, Università di Roma "Tor Vergata": Geometric Flows on Lattices

Daniel Grieser, Universität Oldenburg: Scales, blow-up and quasimode constructions

Tere M-Seara, Universitat Politècnica de Catalunya: Exponentially small splitting of separatrices

Peter Szmolyan, Technische Universität Wien: Advances in Geometric Singular Perturbation Theory

I will explain how the language of manifolds with corners and blow-up can be useful, and indeed is very natural, in dealing with certain types of problems involving different scales, for example for quasimode constructions for ’thin’ domains having non-uniform different scaling behavior. The first part of the course will introduce the basic language and machinery, then some applications will be presented.

In this course we will review a classical problem in perturbation theory. We will consider a system having some hyperbolic manifold (point, periodic orbit, torus etc) with stable and unstable manifolds which coincide and study how they split when perturbing the system. In the regular case, there is a well established  method, known as Melnikov-Poincaré method, which provides an asymptotic formula which measures the difference of the manifolds in a suitable section in terms of the perturbing parameter.

We will study the so-called singular case, where the system is degenerate and this formula is not always valid. We will focus in analytic systems where the splitting is exponentially small in the perturbing parameter. We will show some famous examples where this phenomenon occurs like the Hopf zero-singularity, the restricted three body problem or the problem of Arnold diffusion in the so-called a priori-stable systems. Then we will give some ideas of how can prove a formula which gives the asymptotic measure of the splitting of separatrices in these cases.

Modelling of problems from natural sciences, engineering and life sciences by ordinary differential equations often leads to singular perturbation problems with solutions varying on several widely separated time-scales. The analysis of such systems by methods from dynamical systems theory  - most notably from invariant manifold theory - has become known as geometric singular perturbation theory (GSPT). By the efforts of many people GSPT has been applied successfully in the analysis of an impressive collection of diverse problems. Fenichel theory for normally hyperbolic critical manifolds combined with the blow-up method at non-hyperbolic points is often able to provide remarkably detailed insight into complicated dynamical phenomena. Much of this work has been carried out in the framework of slow-fast systems in standard form, i.e. for systems with an apriori splitting into slow  and fast variables.

More recently GSPT turned out to be useful for systems for which 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, problems depending singularly on more than one parameter, and smooth systems limiting on non-smooth systems as a parameter tends to zero. 

In this course I will survey these developments and explain key features in the context of selected applications.

This is the schedule of the Summer School "Multiscale Phenomena in Geometry and Dynamics" from 22 to 26 July 2019.