Attractors For Several Classes Of Delay Dynamic Systems

Differential equations are very important branch in mathematics, and they are also powerful tools to describe many dynamical systems which evolve over time. From the study of longtime behavior of solutions to the differential equations, we learnt much about the prospective trend of the systems. Along with the development of science and technology, scientists found that, in the real world, the dynamical systems were often affected by delay or random factors. Thus, using differential equations with these factors to simulate will be more close to the real dynamical systems. Therefore, the study of delay dynamical systems and stochastic dynamical systems are important in theory and practical application. In this thesis, we focus on the study of the longtime behavior of the solutions of several types of delay dynamical systems and stochastic delay dynamical systems from the view-point of attractor theory. The outline is as follows.In Chapter1, we briefly introduce the historical background and main research direc-tions of infinite dimensional dynamical systems, stochastic infinite dimensional dynamical systems and our research work. The concepts and propositions of global, pullback and ran-dom attractors are also presented.In Chapter2, we consider the longtime behavior of solutions of a type of retarded lattice dynamical system. At first, applying the priori estimates and tailed estimates, we obtain the existence criterion for the global attractor of this system. Moreover, we analyze the upper semi-continuity with respect to a sequence of finite dimensional approximate system. Finally with numerical experiments, the theoretical results are further illustrated.Chapter3is devoted to considering the asymptotic behavior of solutions of a second order stochastic retarded lattice dynamical system. It shows that, under some dissipative and sublinear growth conditions, such system possesses a compact global random attractor within the set of tempered random bounded sets. In the end, several numerical examples further illustrate the obtained theoretical results.In Chapter4, we investigate the dynamical behavior of a strong damped plate equation containing a delay forcing term. The existence criterion for the pullback attractor of this system is presented. Moreover, there also exists a uniform forward attractor under additional suitable assumptions.In Chapter5, a damped Boussinesq equation containing a delay forcing term is studied. We first prove the existence of a uniformly pullback attracting set by the decomposition technique, and then establish the compactness of the attracting set, which is based on some new estimates of the equicontinuity of the solutions.