Low Dimensional Dynamical Characterization of Partial- Differential Equations

本研究计划的目标是发展计算方法来研究由中高维微分方程系统引起的混沌吸引子的动力学行为。重点是由PDE的控制流体在不稳定和湍流的开始处或稍超过其开始处的惯性流形缩减引起的方程组。PI的兴趣在于描述可能有10-100个自由度的系统的低维动力学。所提出的工作的主要组成部分包括：发展精确计算三维以上吸引子的稳定和不稳定流形的精确数值方法；使用实时彩色图形来改进马鞍轨道和相关流形几何结构的可视化；并利用矢量化算法来开发新型计算机硬件的能力。算法和产生的软件包旨在帮助应用数学家，工程师，物理学家和学生研究非线性常微分方程和偏微分方程系统的动态行为。

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.