Averaging methods for nonlinear systems with a small parameter via reduction and topological degree

When using averaging theory, one usually tries to find the simple zeros of an associated averaged function. When the zero is not simple, the classical averaging theory does not provide information about the periodic solution associated to a non-simple zero. To overcome this difficulty, one way is to perform the high order averaging in case that the system is smooth enough. Here we provide sufficient conditions to keep periodic solutions persisting by the Lyapunov-Schmidt reduction and topological degree theory. Additionally, we perform an application of our result for the forced Galactic dynamics model.