Limit cycles, regularization and period function of piecewise smooth planar systems.

Abstract

The present project has four main lines of research for planar differential systems. First, in a generic situation, we intend to understand the behavior of the regularization of orbits of a piecewise smooth system with four zones, near of a singular point in the discontinuity manifold having an orbit of the "homoclinic type". In a second moment, assuming conditions for the existence of this "homoclinic type" orbit, seek conditions for the existence of limit cycles of the regularized field. Piecewise smooth Hamiltonian systems allow more configurations of centers, so in a third front, we dedicate to the study of aspects and calculation of the period function for this type of systems. Finally, in a fourth approach we will study new versions of Hilbert's sixteenth problem, i.e. to find a lower bound to the number of limit cycles in terms of the number of homogeneous vector fields formed by single monomials involved in polynomial differential systems.