Computational and topological aspects of dynamics

Dynamical systems appear as mathematical models in every area of the natural sciences. In recent years our knowledge of dynamical systems and its links with topology has made tremendous progress. Advances in geometry, knot theory and computational topology are finding applications in the study of dynamical systems. The topological information turns out to have essential implications for observed phenomena such as stationary, periodic and chaotic solutions.

In order to understand the observed long term behaviour in a dynamical system, it is crucial to analyse the invariant dynamics of the system and its dependence on parameters. The reason is that the invariant dynamics, the bounded motions which exist for all (forward and backward) time, often form the global attractor for all motions. Usually these orbits have better properties than arbitrary orbits, exhibiting topological structures which carry essential information and/or converting the study of an infinite dimensional system to a finite dimensional setting.

An important aspect is the computability of topological information. Numerical simulations are a relatively easy way to explore the phenomena exhibited by a particular dynamical systems, but such computations do not reveal any of the reasons for the observed behaviour. On the other hand, topological techniques give a deeper understanding of the underlying structure and capture the behaviour of broad families of systems. To combine these two approaches one must be able to find topological information in a computationally robust manner. Arguments from topology can then be used to justify specific features seen in the numerical simulations.

Advances in applications of topology to dynamics and the rapid increase of computer power make this an ideal opportunity to bring together experts from the fields of topology, dynamics and computational homology to foster collaborations and to develop new links.