Discontinuity-Induced Bifurcations Due to Saturations

Abstract

This paper regards the study of systems of non-smooth differential equations with parameters and saturation discontinuities in the vector field. Once the problem is stated, we define an appropriate solution concept. This regards mainly to the framework of Filippov systems and sliding solutions in the control literature. Second, we consider linear systems with one saturation linear manifold in a two-dimensional state space and deduce several properties that will do analytical bifurcation analysis easier. This allows us to characterize the invariant sets of these systems and thus perform efficient bifurcation analysis. Non-smooth bifurcations due to interaction of an invariant set with a switching-impact-sliding manifold are also known as discontinuity-induced bifurcations. When we consider two-dimensional linear systems depending on parameters trace and determinant of the matrix, and two additional significant parameters, we can compute equilibrium points, pseudo-equilibriums and (smooth and non-smooth) periodic orbits analytically, depending on the parameters.We completely classify the dynamical systems according to the invariant sets that we found. We extend our analysis to two saturation linear manifolds, and generalize our method to several saturation non-linear manifolds and non-linear differential equations through Lie-derivatives framework. This is applied to a simple oscillator circuit which is usually used in control strategies of mechanical systems.