The Impact Feature Of The Trajectory Of Impulsive Differential Systems And Dynamic Research

The phenomena of impulse, being as one of phenomena of transient process, widely exists in the field of modern science and technology and engineering. The mathematical models of many practical problems are often come down to the impulsive differential sys-tem. The most distinguishing feature of impulsive differential systems is able to reflect the laws of things more accurately and deeply by considering the influence of the phenom-ena of transient process. With the development of science and technology, the impulsive differential system has been widely applied into aerospace technology, information sci-ence, biotechnology, economics etc. In view of the significant importance of impulsive differential systems, impulsive differential systems are more and more focused by math-ematics workers, and many great developments have been achieved[1-24,35-40].We call it pulse phenomena, which allowing trajectories meet each surface more than once. Pulse phenomena complicates the investigation of properties of solutions of the impulsive differential systems, especially when it occurs pulse accumulations, which is more difficult than the ones with fixed impulses. Hence, there is a lot of work to do in this field. This paper focus on the research about pulse phenomena and the dynamical analysis of impulsive differential systems with state-dependent impulses. This paper is divided into three parts.In chapter one, we study the impulsive differential system with state-dependent impulses.We get new sufficient theorem about the pulse phenomena by allowing the surfaces are unbounded, which improve previous results. And we get the sufficient conditions to guarantee the existence of two pulse accumulations on the adjacent surfaces. At the same time, we establish a comparison inequality of impulsive differential systems with pulse accumulations.In chapter two, we study the differential functional system with pulse phenomena.With allowing the occurrence of the pulse phenomena, the sufficient conditions of compression practical stability and uniformly practical stability of impulsive differential systems with finite delay in terms of two measurements. In section 3, by using Lyapunov function and Razumikhin technique, we establish a new criteria on compression practical stability. Then we use several Lyapunov functions and Razumikhin technique to guaran-tee the compression practical stability. In section 4, we establish criteria to guarantee the uniformly practical stability of differential functional system by using Lyapunov function and several Lyapunov functions. Finally, examples are given to show the effectiveness of theorem.In chapter three, we study the global exponential stability of the system (?), and we consider cellular neural networks system.In section 3, by using Lyapunov function and Razumikhin technique, we establish a new criteria on the global exponential stability. To verity the practicability of the criteria, then we applied results got in section 3 into the cellular neural networks system.