Global aspects of Filippov Systems via KAM theory

Abstract

In this project, we aim to perform a qualitative study of some global aspects of Filippov systems. Using methods from the qualitative theory of dynamical systems, we shall address problems about the persistence of invariant sets (limit cycles, invariant tori, etc.) and the asymptotic behavior of solutions. More specifically, we shall consider a family F of non-smooth second order differential equations. Firstly, given the nonsmoothness of the equations, we aim to determine transformations on F in order to allow the use of some tools from KAM Theory. In general, the analysis involved in such a study can be very difficult and we have found only a few cases in the literature related to non-smooth systems. These tools guarantee the existence of infinitely many invariant tori and periodic solutions as well as results on the asymptotic behavior of the solutions. Finally, we will perform a Melnilkov analysis to obtain partial results about the existence of periodic orbits of differential equations from F. We shall see that the implementation of this last tool is much simpler than the KAM theory and it can be applied to a wider class of systems. However, the KAM Theory provides stronger and deeper results for specific systems. Therefore, in this project, we shall use, concomitantly, both theories in order to get the results. (AU)