Global infinite-dimensional dynamics
- Research director
- Prof. Dr. Bernold Fiedler
- Dr. Jörg Härterich
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.