Symmetric Hamiltonian systems: Bifurcation theory and numerics

Mechanical systems with symmetry arise for example in molecular dynamics, underwater vehicle dynamics, celestial and spacecraft dynamics and continuum mechanics.For the analysis of the long-time behaviour of such dynamical systems it is crucial to study the bifurcations of its invariant sets as internal parameters like energy and other conserved quantities are varied. The simplest invariant sets of a symmetric dynamical system are equilibria, periodic orbits or orbits which become equilibria or periodic after symmetry reduction, so-called relative equilibria and relative periodic orbits. Whereas the theory of generic symmetry breaking bifurcations of such invariant sets is well-developed for general systems, there are many fewer results on the corresponding theory for symmetric mechanical systems. This is due to the various conservation laws of mechanical systems with symmetry which change the generic behaviour of a dynamical system drastically and therefore have to be taken into account. So far a systematic numerical bifurcation analysis only exists for equilibria and periodic orbits of non-symmetric systems. The aim of this proposal is the parallel development of theoretical and numerical methods for symmetry breaking bifurcations of simple invariant sets of symmetric mechanical systems.The results will applied to various examples of mechanical systems from the areas mentioned above. In particular not only models for the time dynamics of a system, described by ordinary differential equations, will be treated, but the theoretical results will also be extended to models for the dynamics in space and time, ie to partial differential equations and lattice systems.