Symmetry Breaking Hopf Bifurcations in Equations with O(2) Symmetry with Application to the Kuramoto-Sivashinsky Equation

In problems withO(2) symmetry, the Jacobian matrix at nontrivial steady state solutions withD_nsymmetry 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). © 1997 Academic Press.