Bifurcation Structure of the Periodically Forced Relay System

In the present work, we investigate the bifurcation structure associated with the bistability, entrainment of forced oscillations in a relay system with hysteresis. The behavior of such a system is described by a nonautonomous differential equation with discontinuous right hand side. First we discuss the some properties of this equation. Next we present the method for obtaining a first return map from this vector field and show that, depending on the parameter values, such a mapping is a circle diffeomorphism or a gap map. We obtain the equation for the boundary separating the parameter plane into regions where the mapping is smooth invertible or discontinuous. Finally, we perform detailed analytical and numerical bifurcation analysis in order to explain the mechanism of the transition between mode-locked solutions, bistable and irregular dynamics.