Grazing-Induced Dynamics of the Piecewise-Linear Chua’s Circuit

Abstract

In this paper, we consider the piecewise-linear Chua’s circuit, which is well known for its rich variety of bifurcation, chaotic and other nonlinear phenomena. Suitable switching boundaries are introduced based on the piecewise-linear representation of Chua’s diode. In this way, we derive analytical conditions for a grazing bifurcation to occur, when one or two families of periodic orbits have a zero-velocity contact with the switching boundaries. In connection to this phenomenon, we also study the focus-center-limit cycle bifurcation and its implications regarding the system dynamics, from both analytical and numerical points of view. Furthermore, a detailed parametric study of Chua’s circuit is carried out via path-following techniques for nonsmooth dynamical systems, implemented via the continuation software COCO. This study reveals the presence of codimension-one bifurcations of limit cycles, such as those mentioned above, as well as classical (fold and period-doubling) bifurcations. The analysis confirms the presence of coexisting attractors, which are produced by a hysteresis loop induced by the interaction of a fold and a focus-center-limit cycle bifurcation.