Research On Dynamical Behaviour Of Two Important Nonlinear Models

This thesis is devoted to the study of the dynamics for two important nonlinear models: Lienard-type equations and delayed neural networks (DNNs). After a brief description of historical background and research progress in this field, the research work focus mainly on twe parts: The first part discusses the qualitative properties of forced Lienard-type equations as well as some of the related second-order nonautonomous systems. The second part investigates the dynamical behavior of higher-dimensional nonautonomous systems composing of the continuous-time delayed neural networks, which includes some well-known delayed neural networks such as delayed Hopfied neural networks and delayed cellular neural networks (CNNs). The fundamental contributions of these works are summarized in the following three aspects:(1) Global asymptotic behavior of solutions and periodic solutions for forced Lienard-type equations as well as some of the related second- order nonautonomous systems. The main issues are addressed as folllows: Firstly, using the Liapunov function method and the LaSalle invariance principle, some new sufficient conditions as well as the sufficient and necessary conditions of the boundedness and convergence of solutions are given. These results include and improve some important results associated with the same problems in the current literature. Secondly, based on the Mawhin's continuity theorem in coincidence degree theory and the Brousk's theorem, some novel criteron of the existence on periodic solutions are obtained under the more generalized assumptions than before. These results are the improvement and generalization of the main works of Omari, Villari, and Zanolin et, al. Finally, by applying nonlinear functional method and the exponential dichotomy theory, the sufficient conditions of the existence on almost periodic solutions for forced Lienard-type equations are proved, which are the natural extension of the results already obtained by Fink et, al, in particular, a computational error of Fink in SIAM.J.Appl.Math (1974,26(1)26-34) has been corrected.(2) Dynamics of periodic delayed neural networks (PDNNs). The main work is to give some sufficient conditions for the existence and global exponential stability of periodic solutions for PDNNs model, without assuming the smoothness, monotonicity and boundedness of the activation functions. Here the approach is also the Mawhin coincidence degree theory and the Lyapunov functional application, and with help of some important inequalities and matrix-algebraic techniques. Some new and less restrictive yet more general results are consequentlyderived. The new conditions are presented in terms of a nonsingular M-matrix described by the network parameters, the connection matrix and the Lipschitz constant of the activation functions. Furthermore, some examples and simulations are worked out to demonstrate that these conditions are not only simple and practical, but also easier to check and less restrictive than those imposed by earlier similar results.(3) Chaos control and synchronization on Lienard equations and delayed neural networks (DNNs). There are two issues including: First, the robust synchronization problem of some chaotic Lienard equations subject to disturbances is considered, and a practical theory result defiling with analogous problems for a more generalized nonlinear control system is established. Then, the present criteria is applied to two specific chaotic Lienard systems subject to disturbances: the classical van Der Pol oscillator and the corresponding generalized systems, chaos synchronization of such coupled systems are achieved via nonlinear feedback control technique. Moreover, some numerical simulations are worked out to demonstrate the effectiveness of the control approach. Second, using the Lyapunov functional method and Hermitian matrices theory, and base on rigorous mathematical analysis, a simple, less conservative criterion is derived for global synchronization of a networks system composing of identical delayed neur...