On the periodic solutions of switching scalar dynamical systems

In this paper, we investigate the existence and stability of periodic solutions of switching dynamical systems consisting of two sub-equations. We first establish a general criterion to determine the stability of periodic solutions; namely, we derive the conditions under which the periodic solution is locally asymptotically stable, globally asymptotically stable, or unstable. Next, we develop general theorems to count the number of periodic solutions and find the basins of attractions for the periodic solutions and the trivial solution, respectively. As applications, we analyze two biological models in recent literature. Our general theorems not only reproduce the existing results in a unified and simpler manner but also lead to new and complete dynamical results including bistability of the periodic solution and the trivial solution. Numerical examples are also given to illustrate our theoretical results.