A Modified Poincaré Method for the Persistence of Periodic Orbits and Applications

Abstract

This paper is devoted to the persistence of periodic orbits under perturbations in dynamical systems generated by evolutionary equations, which are not smoothing in finite time, but only asymptotically smoothing. When the periodic orbit of the unperturbed system is non-degenerate, we show the existence and uniqueness of a periodic orbit (with a minimal period near the minimal period of the unperturbed problem) by using “modified” Poincare methods. Examples of applications, including the perturbed hyperbolic Navier–Stokes equations, systems of damped wave equations and the system of second grade fluids, are given.