Numerical Analysis of equivariant evolution equations
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.
