Study On The Asymptotical Behavior And Global Dynamical Characteristics Of Infinite Dimensional Dynamical Systems

The study on asymptotic behavior which contains the existence of global attractors with estimation on its radius and Hausdorff dimension is the most important problem in the theory of infinite dynamical system(IDS).Furthermore,according to the results about asymptotic behavior,the modal equation described by ordinary differential equations can be obtained by nonlinear Gakerlin method(NGM)relevant to inertial manifold.Based on the relationship between components of global attractor and some kinds of invariant measures,the global dynamics can be investigated by attaining the components of global attractor numerically.This dissertation concerns with the asymptotic behavior and global dynamics for the vibration of Euler-Bernoulli beam and Von Karman plate as well as the heat conduction.The main research contents and results are as follows.By constructing Ilog inequality and the methods utilized to achieve the Hausdorff dimension estimation of global attractors for autonomous/non-autonomous infinite dynamical system,a approach to estimate the Hausdorff dimension of global random attractors for stochastic dynamic system is proposed.A summary on relationship between some kinds of Invariant measures and components of global attractor,which plays primary role in study on global dynamics,is provided.In order to attain the modal equations associated with infinite dynamical systems quickly,a method rely on the theory of infinite dimensional operator spectrum,NGM and finite element method,realized by COMSOL is argued.Furthermore,a simple example is given to demonstrate the validity of this procedure.Dynamics equation of the Heat conduction process driven by multiplicative white noise with time-varying coefficient can generate a stochastic dynamical system,which can be derived by the existence and uniqueness of solution for the system.The existence of global random attractor and its finite Hasudorff dimensions estimations pertinent to global Lyapunov exponents are derived.Three orders of the modal equations which is employed to get the ingredients of global random attractor numerically is obtained by NGM.Together with the results on asymptotic behaviors,the investigation on global dynamics for the system can be accomplished,which expresses that with the increase of the intensity of linear part in stochastic internal heat source,the dynamics of the system becomes complex.As indicated above,when the finite Hausdorff dimension related to global Lyapunov exponent can be attained exactly,it can be used to study the global dynamics quantitative.Existence and estimation for radius of the uniform attracting set for weakly damping non-autonomous Euler-Bernoulli beam are obtained.The damping of the system is so weak that uniformly asymptotic compactness of the system can not be verified by splitting method which is a typical approach in the investigation on IDS,stabilization estimations which is a primary technique in control theory can be intended to overcome this trouble.Then the system owns Kernel and Kernel Sections.Invoking the splitting methods,the assertion that strongly damping non-autonomous Euler-Bernoulli beam possesses Kernel and Kernel Sections can be validated,moreover,the estimation on Hausdorff dimension of Kernel Sections for above two systems are finite.The global dynamics of the two kinds beams are handled numerically,the results of which reveal that,the increase of longitudinal force of beam or decrease of frequency of axial excitation lead to the global dynamical of the beams becomes complex.On the other hand,the additive white noise delay the occurrence of global bifurcation for the beams.Finally,It is indicated that the crude finite Hausdorff dimension be invoked to study the global dynamics qualitatively.There exists global random attractors for strongly damping Euler-Bernoulli beam driven by additive noise which can induces a random dynamical system(RDS).In addition,the Hausdorff dimension of global random attractor is estimated.the global dynamics is dealt numerically.The assertion tha Hausdorff dimension estimation of global attractor which is excessively conservative can reflect the global dynamic of the system qualitatively can be derived by as indicated above.The existence and estimation for expectation of the radius of global random attractor for clamped Von Karman plates driven by neither additive white noises or multiplicative white noises are implemented.Compared with the previous results,when the system damping and geometric parameters are given,the results of this dissertation indicate that if the boundary value of internal stress is non-zero and the damping is weak,there also exists global random attractors for the two kinds of Von Karman plates.Furthermore,the intensity of additive noise only affect the estimation for expectation of radius of the global random attractors.Nevertheless,both the existence and estimation for expectation of radius of global random attractor are depended on the intensity of the multiplicative noise.In addition,the global dynamics of the two stochastic clamped Von Karman plates are tackled numerically.Moreover,the above studies show that the global dynamics of the system can be qualitatively reflected by the estimation of global random attractors.