Project: DFG research group "Spectral analysis, asymptotic distributions and stochastic dynamics"

Numerical approximation and spectral analysis of infinite-dimensional dynamical systems

Description

In this project we develop and analyze numerical methods for approximating dynamical systems in infinite dimensions. During the past years several novel methods have been proposed in order to compute the longtime behavior of finite dimensional dynamical systems in a direct way, i.e. without longtime integration of single trajectories. One class of methods relates to boundary value problems in time by which certain special solutions such as equilibria, periodic, and homoclinic orbits can be approximated. The second class comprises methods for covering attractors by systems of boxes that can be refined iteratively by using a multitude of short trajectories.

In both cases the direct methods are followed by a spectral analysis. In the first case one investigates the spectrum of a linearization about the orbit in order to obtain information about its stability while in the second case one computes eigenvalues and eigenmeasures of transfer operators (Perron Frobenius) in order to specify the dynamics that remains on the attractor. One of the goals of this project is to contribute to the extension of these methods and their analysis to certain infinite dimensional systems, such as parabolic partial differential equations and lattice dynamical systems. The main topic of the theory will be to investigate the effects of time and space discretizations, in particular the transition from an infinite to a finite spatial domain.

A specific part of this general range of problems are continuous time systems on infinite lattices (lattice dynamical systems). They appear in many applications of statistical mechanics, neural networks and chemical reaction kinetics. Travelling pulses and fronts ar prominent phenomena in the longtime behavior of such systems. Numerical methods that restrict to finite domains and use asymptotic boundary conditions will be analyzed and it is also intended to investigate structural changes in the spectra when the mesh refines towards an underlying partial differential equation. The numerical methods for computing clusters of eigenvalues will also be used in order to analyze the behavior of spectra on irregular grids such as prefractals.

Numerical methods that have been developed to cover attractors in finite dimensional systems will be adapted to include the translation invariance of an infinite lattice and to incorporate appropriate asymptotic boundary conditions. Finally, it is of interest to analyze the behavior of spectra and eigenmeasures of transfer operators under this approximation.

Members

Preprints