case, the degeneracy is dealt with by splitting the vector field into two parts, one tangent to the group orbit and the other
In problems with O(2) symmetry, the Jacobian matrix at nontrivial steady state solutions with D n symmetry always has a zero eigen-normal to it. A standard bifurcation analysis is then pervalue due to the group orbit of solutions. We consider bifurcations formed on the normal vector field and the results are interwhich occur when complex eigenvalues also cross the imaginary preted for the whole vector field. He considers the Kuraaxis and develop a numerical method which involves the addition moto-Sivashinsky equation with O(2) symmetry as an of a new variable, namely the velocity of solutions drifting round example but does not present any new numerical results.
the group orbit, and another equation, which has the form of a phase condition for isolating one solution on the group orbit. The Landsberg and Knobloch [22] also considered this probbifurcating branch has a particular type of spatio-temporal symmelem and showed that at a Hopf bifurcation on the steady try which can be broken in a further bifurcation which gives rise to state solution, so-called direction reversing travelling wave modulated travelling wave solutions which drift around the group (DRTW) solutions bifurcate. They transformed the proborbit. Multiple Hopf bifurcations are also considered. The methods lem into canonical coordinates [8] so that one equation, derived are applied to the Kuramoto-Sivashinsky equation and we give results at two different bifurcations, one of which is a multiple associated with the drift around the group orbit, decouples Hopf bifurcation. Our results give insight into the numerical results from the others thus separating out the degeneracy. Procof Hyman, Nicolaenko, and Zaleski (Physica D 23, 265, 1986). ᮊ 1997 tor and Weiss [24] and Matthews, Proctor, Rucklidge, and Academic Press
Weiss [25] have also considered this problem in the context of nonlinear magnetoconvection. They refer to the bifurcating periodic solutions as pulsating waves. They also