# Project: DAAD research exchange program with Hungary

## Qualitative theory of numerical methods for evolution equations in infinite dimensions

#### Description

This research project belongs to the field of

**Numerical Dynamics**, an important topic in the general mathematical analysis of dynamical systems and their applications. The purpose of this field is to compare the qualitative properties of continuous time dynamical systems (i.e. the longtime behavior of ordinary, delay or partial differential equations) with their discrete-time counterparts which are generated by numerical methods. For the proper interpretation of computer simulations it is a vital question if special configurations of dynamical systems (such as equilibria, periodic and homoclinic orbits, invariant manifolds and bifurcations) are preserved or destroyed by the discretization process. Numerical dynamics is a relatively new subject and its development was largely stimulated by the general availability of computers. During the past 15-20 years most of the basic qualitative results for ordinary differential equations have found their analogues in numerical dynamics. Meanwhile it has become clear that the frontiers of research have passed from finite to infinite-dimensional systems, i.e. from ordinary to partial and delay differential equations. It is the main intention of this project to get young researchers from both sides

*from Hungary and from Germany* involved in this process. By close cooperation with the senior participants they should be able to keep pace with the general development and to exchange ideas about theory and also software applications. The aim of the research project is to investigate the qualitative numerics of

- reaction-diffusion equations,
- delay equations.

These are the two types of infinite-dimensional dynamical systems which are nearest to ordinary differential equations and which show behaviour resembling the finite dimensional case. A particular emphasis is put on discretization effects for the following topics

- higher order smoothness of invariant manifolds,
- bifurcation problems (saddle nodes, Hopf points, period doublings, homoclinic orbits),
- parabolic systems with applications in chemical engineering.

#### Members

#### Preprints