Preprint des Projektes: SFB 701: Spektrale Strukturen und Topologische Methoden in der Mathematik - Projekt B3

Numerische Analyse äquivarianter Evolutionsgleichungen

07-047 Wolf-Jürgen Beyn, Vera Thümmler.
Phase conditions, symmetries, and PDE continuation

Integral phase conditions were first suggested by E.J. Doedel as an efficient tool for computing periodic orbits in dynamical systems. In general, hase conditions help in eliminating continuous symmetries as well as in educing the effort for adaptive meshes during continuation. In this paper we discuss the usefulness of phase conditions for the numerical analysisof finite- and infinite-dimensional dynamical systems that have continuous symmetries. The general approach (called the freezing method) will be presented in an abstract framework for evolution equations that are equivariant with respect to the action of a (not necessarily compact) Lie group. We show particular applications of phase conditions to periodic, heteroclinic and homoclinic orbits in ODEs, to relative equilibria and relative periodic orbits in PDEs as well as to time integration of equivariant PDEs.