Oscillating waves arising from O(2) symmetry

In the problems with O(2) symmetry, the Jacobian matrix at the branch of nontrivial ℤ2 symmetric steady-state solutions always has a zero eigenvalue due to the group orbit of solutions. Hopf bifurcation has been considered where a pair of complex eigenvalues also crosses the imaginary axis. Canonical coordinates have been used to remove the degeneracy of the system. The bifurcating branch has a particular type of spatio-temporal symmetry which can be broken in a further bifurcation giving rise to modulated travelling wave solutions which drift round the group orbit.