The periodic solutions of the impulsive state feedback dynamical system

Abstract

This work reviews some recent advances on the periodic solution of the semi-continuous dynamical system, which consists of two parts: the stability of periodic solution, the homoclinic and heteroclinic bifurcations. In the first part, the order-1 periodic solution is classified into three types at first. Then for type 1 periodic solution, by means of square approximation and a series of switched systems, the periodic solution is approximated by a series of continuous hybrid limit cycles. Hence, a general stability criteria are obtained by the method of successor function similar to the analysis in the ordinary differential equation. In the second part, the homoclinic and heteroclinic cycles are found for some specific parameter value in the prey-predator system. When the parameter varies, the cycles disappear and the system bifurcates an unique order-1 periodic solution. The geometry theory and the successor function are applied to obtain these bifurcations. Finally, we discuss some possible future trends in the periodic solution of the semi-continuous dynamical systems.