Minicurso ministrado no Summer School on Qualitative Theory of Piecewise Ordinary Differential Equations.
Resumo: The averaging method is an important and celebrated method for dealing with nonlinear oscillating systems in the presence of small perturbations. It is mainly concerned with providing long-time asymptotic estimates for solutions of perturbed non-autonomous differential equations. This method has also been extensively employed in the study of periodic solutions.
We start by discussing the classical averaging method for smooth systems and its relation with a Melnikov-like procedure. Then, we shall explore results for obtaining necessary conditions for the existence of periodic solutions for smooth systems. The generalizations of such results to the nonsmooth continuous and discontinuous contexts will be fairly discussed. Finally, if there is time, we may also approach further topics, such as bifurcations from non-degenerate families of periodic solutions and torus bifurcation.
