Periodic Orbits of n-Dimensional Impulsive Switched Systems

In this paper, we generalize the index of impulsive periodic orbits defined in planar systems to n-dimensional impulsive switched systems. The moving Poincaré section is applied, which extends the classical Poincaré section and makes it possible to always preserve the transversal property at the corresponding time t with the help of reparametrization. Further, we combine it with the implicit function theorem and linear approximation to rewrite the original n-dimensional differential equation, which contributes to the definitions of indexes of impulsive periodic orbits of arbitrary finite order m. Based on these indexes, we further obtain sufficient conditions to judge the Zhukovskiǐ quasi-stability, which permits a time lag and focuses on the study of approximate synchronization between orbits with the existence of impulsive switchings. At last, an example of harvesting fishery model with mutual impacts of infection and abrupt harvesting firstly is presented and some sufficient criteria based on model parameters are obtained to illustrate our results.