Configuration of Ten Limit Cycles in a Class of Cubic Switching Systems

The bifurcation of limit cycles is an important part in the study of switching systems. The investigation of limit cycles includes the number and configuration, which are related to Hilbert’s 16th problem. Most researchers studied the number of limit cycles, and only few works focused on the configuration of limit cycles. In this paper, we develop a general method to determine the configuration of limit cycles based on the Lyapunov constants. To show our method by an example, we study a class of cubic switching systems, which has three equilibria: (0, 0) and (±1, 0), and compute the Lyapunov constants based on Poincaré return map, then find at least 10 small-amplitude limit cycles that bifurcate around (1, 0) or (−1, 0). Using our method, we determine the location distribution of these ten limit cycles.