ABSTRACT FORCED SYMMETRY-BREAKING AND FORCED FREQUENCY LOCKING OF MODULATED WAVES

We consider abstract forced symmetry breaking problems of the type F(x , lambda) = y. It Is supposed that for all lambda the maps F(. lambda) are equivariant with respect to the action of a compact Lie group. th at F(x(0), lambda(0)) = 0 and, hence, that F(x, lambda(0)) = 0 for all elements x of the group orbit O(x(0)) of x(0). We look for solutions I which bifurcate from the solution family O(x(0)) as lambda and y mov e away from lambda(0) and zero, respectively. Especially, we describe the number of different solutions x (for fixed control parameters lamb da and y), their dynamic stability and their asymptotic behavior for y tending to zero. Further, generalizations are given to problems of th e type F(x, lambda) = y(x, lambda). Finally. our results are applied t o a forced frequency locking problem of the type (x) over dot (t) = f( x(t), lambda)-y(t). Here it is supposed that the vector Fields f(., la mbda) are S-1-equivariant, that the unperturbed equation (x) over dot = S(x, lambda(0)) has an orbitally stable modulated wave solution and that the forcing y(t) is a modulated wave. (C) 1998 Academic Press.