Low Dimensional Dynamical Characterization of Partial- Differential Equations

该研究项目的目标是开发计算 混沌动力学行为的研究方法 吸引子的微分方程系统， 适度的高维度。 重点是方程组 由于偏微分方程控制流体的惯性流形简化而产生的 在不稳定性开始时或稍微超过不稳定性开始时流动， 急流. PI的兴趣在于描述低 潜在具有10-100维的系统的维动力学 自由度。 拟议工作的主要组成部分 包括：发展精确的数值方法， 稳定和不稳定吸引子流形的精确计算 在三维以上;使用实时彩色图形， 改进鞍轨道几何结构的可视化 和相关的流形;以及使用矢量化算法 开发新电脑硬件的能力 的算法和 由此产生的软件包旨在帮助应用 数学家、工程师、物理学家和学生 非线性常、偏微分方程系统的动力学行为 微分方程

The goal of this research project is to develop computational methods for investigating the dynamical behavior of chaotic attractors resulting from systems of differential equations of moderately high dimension. The emphasis is on systems of equations arising from inertial manifold reductions of PDE's governing fluid flows at and somewhat beyond the onset of instabilities and turbulence. The PI's interest is in characterizing the low dimensional dynamics of systems which potentially have 10-100 degrees of freedom. The principal components of the proposed work include: the development of accurate numerical methods for the precise calculation of stable and unstable manifolds of attractors in more than three dimensions; use of real-time color graphics for improved visualization of the geometric structure of saddle orbits and the associated manifolds; and the use of vectorized algorithms to exploit the power of new computer hardware. The algorithms and resulting software packages are intended to assist applied mathematicians, engineers, physicists, and students investigating the dynamical behavior of systems of nonlinear ordinary and partial differential equations.