From Topological to Stochastic Techniques in Dynamical Systems

Dynamical systems are mathematical models that appear in many areas of the natural sciences. They describe the evolution of chemical reactions, the motion of satellites around the earth and the outbreak of diseases. Their solutions determine the shapes and speeds of signals in optical fibers, the size of convective rolls in the atmosphere and the arrangement of atoms in a crystal.

In view of the large variety of these natural phenomena, it comes as no surprise that the dynamical models that are used to describe them take many different forms. They range from simple and classical low-dimensional maps and ordinary differential equations to complex networks, partial differential equations, coupled map lattices, delay equations, stochastic differential equations, hybrid systems and many more. And even so, no such model is ever a perfect description of the world around us, which explains why one of the main theoretical goals of this field has always been to understand which dynamical phenomena are robust under changes in the model or even completely independent of it.

In fact, much of our knowledge of the dynamics of maps and ordinary differential equations was formed in the 1960s and 1970s. On the other hand, recent years have seen an enormous progress in our understanding of the qualitative behaviour of the more complicated and high-dimensional models. Examples of these developments include the use of various topological methods in the study of dynamical systems, the theory of existence and stability of nonlinear waves, the birth of a theory of coupled cell network dynamical systems, the usage of variational methods in statistical mechanics and material science, a rush of developments in the ergodic theory of deterministic systems and our recent understanding of random and stochastic maps, ODEs and PDEs.

Many of these recent developments are happening quickly now and much of the theory behind them is technical and hard to access for a relative outsider. Topological and stochastic methods are rapidly progressing perspectives in dynamical systems; rigorous computational techniques, equivariant bifurcation theory and network dynamics are other examples of prolific areas. It is our belief that scientists working in these different fields, including ourselves, are often not keeping up with these developments. As a consequence, we think that many of the techniques that have recently been developed are being far from fully exploited. Researchers in dynamical systems could benefit greatly from a better understanding of each others work.

This is the reason for this workshop at the Lorentz Centre: we invited researchers in these quite diverse modern areas of dynamical systems to give introductory lectures on their respective fields of expertise. The aim is to provide their audience with an overview of important developments and open problems. Indeed, the focus of the lectures will not be on technicalities but rather on important questions, leading ideas and crucial insights. We hope that this will broaden the horizon of the workshop participants.