Abstract Forced Symmetry Breaking and Forced Frequency Locking of Modulated Waves

We consider abstract forced symmetry breaking problems of the typeF(x, λ)=y. It is supposed that for allλthe mapsF(•, λ) are equivariant with respect to the action of a compact Lie group, thatF(x0, λ0)=0 and, hence, thatF(x, λ0)=0 for all elementsxof the group orbit (x0) ofx0. We look for solutionsxwhich bifurcate from the solution family (x0) asλandymove away fromλ0and zero, respectively. Especially, we describe the number of different solutionsx(for fixed control parametersλandy), their dynamic stability and their asymptotic behavior forytending to zero. Further, generalizations are given to problems of the typeF(x, λ)=y(x, λ). Finally, our results are applied to a forced frequency locking problem of the typex(t)=f(x(t), λ)−y(t). Here it is supposed that the vector fieldsf(•, λ) areS1-equivariant, that the unperturbed equationx=f(x, λ0) has an orbitally stable modulated wave solution and that the forcingy(t) is a modulated wave.