Low Dimensional Dynamical Characterization of Partial- Differential Equations

这个研究项目的目标是发展计算方法来研究由中高维微分方程组产生的混沌吸引子的动力学行为。重点是由偏微分方程组的惯性流形简化所产生的方程组，这些方程是在不稳定和湍流开始时或略高于不稳定和湍流开始时的流体流动。PI的兴趣在于描述可能具有10-100个自由度的系统的低维动力学。拟议工作的主要内容包括：开发精确计算三维以上稳定和不稳定吸引子流形的精确数值方法；使用实时彩色图形改进鞍形轨道及其相关流形的几何结构的可视化；以及使用矢量化算法来开发新的计算机硬件的能力。算法和生成的软件包旨在帮助应用数学家、工程师、物理学家和研究非线性常微分方程组和偏微分方程组的动力学行为的学生。

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.