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.
Highlights
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
We aim to review the advances of the periodic solutions of of semi-continuous dynamical systems since 2010, the existence and stability of periodic solution and the homoclinic and heteroclinic bifurcations [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]
This paper reviews some advances on the stability and bifurcations of the semi-continuous dynamical systems since 2010
Summary
Impulsive control methods have many important applications in various fields such as biology, engineering, medicine etc[1]. In the references [26, 25, 35, 34], the authors applied this method to obtain some stability results for the particular semi-continuous dynamical systems Based on these results, the authors in [44] classified the order-1 periodic solution into three types at first, and presented a convenient and general stability criteria of the convex periodic solution by square approximation and a series of switched systems.
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