BIFURCATION TO ROTATING WAVES IN EQUATIONS WITH O(2)-SYMMETRY

This paper considers the detection and calculation of bifurcations from steady-state solutions to rotating (or travelling) wave solutions of the time-dependent nonlinear problem dx/dt+g(x,lambda) = 0, where g is equivariant with respect to an action of the group O(2). It follows from the equivariance condition that at every nontrivial, steady-state solution (x,lambda), g(x)(x,lambda) has a zero eigenvalue with a one-dimensional null space. It is shown that when a second real eigenvalue passes through zero as lambda varies, and the null space of g(x)(x,lambda) remains one-dimensional, then bifurcation to rotating waves occurs subject to a nondegeneracy condition. It is known that rotating wave solutions satisfy a "steady-state" equation. A phase condition is added and a reflectional symmetry that is broken when bifurcation to rotating waves occurs is defined. Thus standard, steady-state, symmetry-breaking bifurcation theory can be used to analyse this type of bifurcation. An extended system for the direct computation of the bifurcation point is introduced and analysed, and possible modes of bifurcation from rotating waves are also considered. The theory is illustrated by numerical results for the Kuramoto-Sivashinsky equation.