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 stea dy state solutions with D-n symmetry 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 n umerical 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 so lution on the group orbit. The bifurcating branch has a particular typ e of spatio-temporal symmetry which can be broken in a further bifurca tion which gives rise to modulated travelling wave solutions which dri ft around the group orbit. Multiple Hopf bifurcations are also conside red, The methods derived are applied to the Kuramoto-Sivashinsky equat ion and we give results at two different bifurcations, one of which is a multiple Hopf bifurcation. Our results give insight into the numeri cal results of Hyman, Nicolaenko, and Zaleski (Physica D 23, 265, 1386 ). (C) 1997 Academic Press.