On Qualitative Research For Several Kinds Of Differential Dynamic Systems

In this dissertation, the qualitative research for several kinds of differential dynamic systems are investigated by employing the continuation theorem of Mawhin's coincidence degree theory, M-matrix, topological degree theory, Liapunov functional method and some inequality techniques. Some new results are obtained, which extend and improve some existing results reported in the literature. These results possess important leading significance in the theory and applications.In chapter 1, the background and applications of what we study in this paper are simply outlined. The contents and main results are also given.In chapter 2, by using the method of analogue and Liapunov's second method, the global asymptotic stability of a class of fourth-order nonlinear system is studied. Several sufficient con-ditions of global asymptotic stability are obtained by constructing a suitable Liapunov function. The nature of infinity of Liapunov function is replaced by the boundary of positive half trajectories of the system in the proof of the theorem. The stronger conditions on Liapunov function trend to infinite is removed. Some important results are obtained and some nonlinear systems in prior works are extended.In chapter 3, on the condition that linear impulsive dynamic systems are mostly researched in existing literature, and the influence of nonlinear impulsive is less studied. In this chapter, the influence of nonlinear impulsive for a kind of cellular neural networks dynamic system with bounded delays and distributed delays is discussed by using the coincidence degree theory, nonsingular M-matrix theory, Liapunov functional method and combine with some analysis techniques. The sufficient conditions of global exponential stability of the equilibrium are obtained.To the best of our knowledge, there is few paper considering the exponential synchronization for the reaction-diffusion neural networks when the status variable on the boundary is zero. In chapter 4, the exponential synchronization for the drive-response system of the reaction-diffusion neural networks with bounded delays and distributed delays is investigated under the condition that the status variable on the boundary is zero. By using some inequality techniques and constructing suitable Lyapunov functional, some sufficient conditions are obtained to ensure the exponential synchronization of the drive-response neural networks system. In chapter 5, by employing the continuation theorem of Mawhin's coincidence degree theory, Liapunov functional and some inequality techniques, some sufficient conditions are obtained to ensure the existence and the global exponential stability of the periodic solution to the BAM-type Cohen-Grossberg neural network involving distributed delays and bounded delays. Milder constraints are imposed on BAM-type Cohen-Grossberg neural network dynamic system.