Nonlinear behaviors as well as the mechanism in a piecewise-linear dynamical system with two time scales

The main purpose of the work was to explore the bursting oscillations as well as the mechanism in a periodically excited piecewise-linear system with an order gap between the exciting frequency and the natural frequency. Based on the typical Chua’s circuit, a periodically excited model is established, in which the nonlinear characteristics of the resistor are expressed in terms of a continuous function with multiple piecewise-linear segments. By analyzing the nominal equilibrium orbits of the linear subsystems in different regions divided by the break points, critical conditions corresponding to regular and non-smooth bifurcations are derived. Two typical cases in which the trajectories pass across different numbers of non-smooth boundaries are investigated, resulting in different forms of bursting oscillations. It is pointed that not only the properties of the nominal equilibrium orbits in different regions but also the non-smooth boundaries may influence the bursting attractors. Furthermore, unlike the connections between quiescent states (QSs) and spiking states (SPs) via regular bifurcations in smooth vector fields, for the bursting oscillations in piecewise-linear systems, the transitions between QSs and SPs may be caused by non-smooth bifurcations or the abrupt alternations between two different subsystems on both sides of the non-smooth boundaries.