Solutions of slowly and periodically varying differential equations. Applications to some classical examples

We study a singularly perturbed second-order differential equation describing slowly and periodically varying hamiltonian system. Typical dynamics governed by this type of system are, for example, equations Duffing or the “shallow water sloshing” problem. Using symmetries singular perturbation tools, we describe dynamics, splitting phase space in regions where motion is oscillatory others it unbounded, each kind regions. Finally establish existence periodic solutions give structure these term response curves. In particular, our results extend complete ones stated [1], [5], [7] answer to some open questions within. To illustrate results, conclude work numerical classical examples.