DFG Project

Global infinite-dimensional dynamics

Research director

Prof. Dr. Bernold Fiedler

Members

Dr. Jörg Härterich

Summary

There are many examples showing that the long-time behaviour of dynamical systems is largely affected or even governed by global objects such as homoclinic orbits, invariant manifolds or attractors. Also, global features like symmetries or reversibility have a major influence on the dynamics. The work has been concentrated on the following three aspects:

1st) Dimension reduction

Dimension reduction is an important tool since often the dynamics of a system is governed by the dynamics on small and sometimes even finitely-dimensional set. Major progress has been made in the description of the compact attractors of scalar reaction-diffusion-drift equations by Fiedler and Rocha. Up to so-called connection equivalence the attractors can be identified from information on the equilibrium states only. Also, if one is interested in globally bounded solutions a reduction onto a lower-dimensional manifold (e.g. near a homoclinic orbit) is possible.

2nd) Homoclinic and heteroclinic bifurcations

Different work has lead to a better understanding of some codimension two homoclinic bifurcation. These bifurcation are sometimes accompanied by complicated dynamics like Smale horseshoes, period doubling cascades or further homoclinic orbits while in other cases none of these features occurs. The distinction between these two cases ("chaotic" and "tame") is the basis of a pathfollowing approach to put together the different examples that have been studied by now.

3rd) Symmetry and reversibility

In the project some work has also been done on systems that possess certain symmetries. Especially the behaviour of periodic and homoclinic orbits in reversible systems has been studied.