Hopf bifurcation in problems with symmetries: Method of phase condition

Abstract:
Hopf bifurcation on a branch of non-trivial steady states in problems with $O(2)$ symmetry is considered. A method, which involves the addition of an equation in the form of phase condition, for dealing with this bifurcation is introduced. The method is applied to a problem in $C^3$ and the oscillating waves are obtained.