# Preprint des Projektes: DFG Schwerpunkt Programm: Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme

## Verbindungsorbits in hochdimensionalen dynamischen Systemen

30/99 Wolf-Jürgen Beyn, Vera Thümmler, Winfried Kleß.
A continuation framework for invariant subspaces and its application to traveling waves

We present a continuation method for low-dimensional invariant subspaces of a parametrized family of large and sparse matrices. Such matrices typically occur when linearizing about branches of steady states in reaction-diffusion equations. Our continuation method provides bases of the invariant subspaces depending smoothly on the parameter. From these we can compute the corresponding eigenvalues efficiently. The predictor and the corrector step are reduced to solving bordered matrix equations of Sylvester type. For these equations we develop a bordered version of the Bartels-Stewart algorithm. The numerical techniques are used to study the stability problem for traveling waves in two examples: the Ginzburg-Landau and the FitzHugh-Nagumo system. In these cases there always exists a simple or multiple eigenvalue zero while the remaining eigenvalues determine the stability. We discuss the difficulties of separating these critical eigenvalues from clusters of eigenvalues that are generated by the essential spectrum of the continous problem.