An introduction to piecewise smooth dynamics

Abstract

In this project, we will make an introductory study of the so-called smooth piecewise dynamical systems (or Filippov systems), which are systems of ordinary differential equations whose vector field does not have smoothness properties. We will study several mathematical concepts and ideas involving such systems, for example, the definition of local trajectory, the sliding vector field (or Filippov field), typical singularities, topological equivalence, periodic orbits, separatrices and the well-known regularization method. In addition, returning to the theory of smooth systems, we will study the concept of structural stability and Peixoto's theorem, as well as the simplest types of bifurcations that can occur in one-parameter families of smooth vector fields.