A general averaging method for affine periodic solutions

We consider the persistence of affine periodic solutions for perturbed affine periodic systems. Such (Q, T)-affine periodic solutions have the form x(t+ T) = Qx(t) for all t∈ R, where T> 0 is fixed and Qis a nonsingular matrix. These are a kind of spatiotemporal symmetric solutions, e.g. spiral waves. We give the averaging method for the existence of affine periodic solutions in two situations: one in which the initial values of the affine periodic solutions of the unperturbed system form a manifold, and another that does not rely on the structure of the initial values of the unperturbed system’s affine periodic solutions. The transversal condition is determined using the Brouwer degree. We also present a higher order averaging method for general degenerate systems by means of the Brouwer degree and a Lyapunov-Schmidt reduction.