Study On The Existence And Stability Of Periodic Solutions For Impulsive Differential Systems

This Ph.D. thesis is divided into five chapters and main contents are as follows:In Chapter 1, a survey to the developments of the existence and stability of periodic solutions of impulsive differential equations, impulsive predator-prey system and impulsive neural networks is given. Then the background of problems, the main results of this dissertation are introduced and some preliminaries are also summarized.In Chapter 2, firstly a class of one predator n-patch-prey model with diffu-sion, impulsive and delays are established. By using Mawhin continuation theorem, Lyapunov functional method and some analysis techniques, the sufficient condi-tions ensuring the existence of positive periodic solutions and global attractivity are obtained in section 2.1. Secondly, by using impulsive comparison theorem, an important lemma and some analysis techniques, the sufficient conditions of the existence of boundary periodic solution, global attractivity and the permanence for a predator-prey system with impulsive diffusion and functional response are obtained in section 2.2.In Chapter 3, an impulsive predator-prey system with Ivlev-type functional response and disease in the prey is considered. By using Floquet theory and Lyapunov functional, the existence and global attractivity of the prey-extinction periodic solution and the permanence of the system are studied. Finally, the complexity of the dynamical behaviors for the predator-prey system is showed by numerical analysis. It is important and valuable for IPM control.In Chapter 4, by using Mawhin continuation theorem, an important lemma and some analysis techniques, the sufficient conditions ensuring the existence of the periodic solution, global attractivity and the permanence of a generalized delayed differential system with impulses are obtained. The main results generalize or improve the previously known results.In Chapter 5, a class of neural networks with delays and impulses is considered in section 5.1. By using Young inequality and constructing Lyapunov functional, Sufficient conditions ensuring the existence and exponential stability of periodic solutions are obtained. Next, a class of BAM neural networks with time-varying delays and impulses is considered in section 5.2. The existence and stability of periodic solution are studied by using continuation theorem, M-matrix theory and Lyapunov functional. Finally, some applications and examples are given to show the effectiveness of the main results.