The Fields Institute for Research in Mathematical Sciences

This paper is devoted to the study of the persistence of periodic solutions under perturbations in dynamical systems generated by evolutionary equations, which are not smoothing in finite time, but only asymptotically smoothing. Assuming that the periodic solution of the unperturbed system is non-degenerate, we want to prove the existence and uniqueness of a periodic solution for the perturbed equation in the neighbourhood of the unperturbed solution (with a period near the period of the periodic solution of the unperturbed problem). We review some methods of proofs, used in the case of systems of ordinary differential equations, and discuss their extensions to the infinite-dimensional case. Mathematics Subject Classification (2010): Primary 35B10, 35B25, 37L50, 37L05; Secondary 35Q30, 35Q35