Symmetry breaking Hopf Bifurcations in equations with O(2) symmetry with application to the Kuramoto–Sivashinsky equation

In problems with O(2) symmetry, the Jacobian matrix at nontrivial steady state solutions with Dnsymmetry always has a zero eigenvalue due to the group orbit of solutions. We consider bifurcations which occur when complex eigenvalues also cross the imaginary axis and develop a numerical method which involves the addition of a new variable, namely the velocity of solutions drifting round the group orbit, and another equation, which has the form of a phase condition for isolating one solution on the group orbit. The bifurcating branch has a particular type of spatio-temporal symmetry which can be broken in a further bifurcation which gives rise to modulated travelling wave solutions which drift around the group orbit. Multiple Hopf bifurcations are also considered. The methods derived are applied to the Kuramoto–Sivashinsky equation and we give results at two different bifurcations, one of which is a multiple Hopf bifurcation. Our results give insight into the numerical results of Hyman, Nicolaenko, and Zaleski (Physica D23,265, 1986).