The Reduction of Equivariant Dynamics to the Orbit Space for Compact Group Actions

Symmetry introduces degeneracies in dynamical systems, as well as in bifurcation problems. An “obvious” idea in order to remove these degeneracies is to project the dynamics onto the quotient space obtained by identifying points in phase space which lie in the same group orbits (the so-called orbit space). Unfortunately, several difficulties arise when one tries to implement this idea. First, the orbit space is not, in general a manifold. Second, how does one explicitely realize the orbit space, and how does one compute and analyze the projected dynamics? In this paper I will describe the methods which have been developped in order to answer these questions, and I will show on three examples how they apply. We shall see that, although not always suitable to treat equivariant dynamics, these methods sometimes lead to insightful reductions.