Infinite Dimensional Dynamic Systems Of Some Nonlinear Elastic Structures

Nonlinear elastic structure is one of the most important research content in solid mechanics,and is the main research object in nonlinear dynamics.The main content of nonlinear dynamics is to correctly understand the complicated dynamic phenomena such as bifurcation,chaos,soliton,fractal,etc.In this paper,we mainly investigate chaos dynamics,which sets up bridge between determinism and probability theory,and attractor is the best tool to study chaotic dynamics.In this paper,based on a lot of research on the dynamic model of solid structures,we generalize the elastic structure of rod and beam in the aspects of increasing complex term,variable coefficient damping,dimension,etc,and consider four kinds of infinite-dimensional dynamical systems.The main results are as follows:1.We consider a class of elastic structure with nonlinear damping and nonlinear external force terms.By using classical operator semigroup theory,we prove the exis-tence and uniqueness of the solution of the system.The existence of global attractor for the infinite dimensional dynamical system is proved by the classical operator semigroup decomposition method.The system is (?)2.We consider a class of dissipative Sine-Gordon-Kirchhoff elastic structure with nonlinear damping and external term.By using the methods of Galerkin approximation and prior estimates,we obtain the existence of global attractors.Firstly,we prove that the system has a unique global solution through prior estimation.Secondly,we prove that the system has bounded absorbing set and smooth operator semigroup.Lastly,we obtain the global attractor of the system.The system we investigate is:(?)The research of global attractor is ideal,and its research work has been improved.Many scholars begin to study more complicated cases of elastic structure,for exam-ple,the uniform attractor-related to time in nonlinear partial differential equation,or random attractor-partial differential equation including stochastic term such as the white noise term,or changing the boundary conditions and so on.We study a more complicated cases on the basis of the above in the next part.3.We mainly discuss the existence of stochastic attractors for nonautonomous stochastic wave equations with white noise,the nonlinear damping has a critical cubic growth rate.By proving the pull back asymptotical compactness of random variables in random dynamical systems,we prove the existence of random attractors.The equation is defined in the bounded region u?R3,and with smooth boundary (?)u:(?)4.We consider the solution of asymptotic property of the autonomous stochastic wave equation with white noise and Neumann boundary condition,the existence of random attractors is obtained by proving the compactness of the absoring set.The system is as follows:(?)...