基于无穷维Arnold随机动力系统理论的多连接梁动力学行为全局分析

渐近行为与全局动特性是动力学行为全局分析的重要研究内容，前者侧重于定性分析，后者侧重于定量研究。本课题以多连接梁随机振动为研究背景，主要研究受到横向/纵向白噪声激励，横向白噪声激励和纵向周期激励的多连接梁的渐近行为与全局动特性。通过数学建模得到系统由多个偏微分方程和连接条件描述的动力学方程，通过算子半群验证其可以生成随机流，从而可将其纳入无穷维Arnold随机动力系统研究范畴。利用全局随机吸引子理论与镇定性估计分析系统的渐近行为，主要包括：系统全局随机吸引子的存在性，吸引子半径的期望估计与Hausdorff维数估计。由Galerkin型方法得到系统的运动方程，通过Hausdorff维数估计刻画全局Lyapunov指数，结合随机细分算法和全局随机吸引子组成部分与随机不变测度的支撑关系研究系统全局随机稳定性与全局D分岔等动特性。最后，基于以上结果分析“随机跳变”等工程实际现象产生的机理。

This proposal plans to consider asymptotical behavior and global dynamics of the following three systems, the multi-link beam driven by additive or multiplicative white noises, the multi-link beam driven by additive and longitudinal periodic excitation. The govern equations which are described by partial differential equations for the systems mentioned above can be got by the mathematical modeling, furthermore, by applying the semigroup theory, it can be proved that the systems can generate random dynamical systems established by Arnold. Invoking global random attractors theory and estimation of stabilization for the systems to handle the investigation on asymptotical behavior of the systems which include the existence of global random attractors, the estimation of expectation for the radius of the global random attractors and the Hausdorff dimension estimation. In light of the Galerkin methods, the equations of motion for the systems can be got, by global Lyapunov exponent which can be described by Hausdorff dimension estimation, together with random subdivisionalgorithm and the relationship between global random attractors and random invariant measure, the global dynamics include global stability and D Bifurcations can be analyzed. Finally, the mechanism of some engineering phenomenon include random snap-through can be obtained by the theoretical results.