Computational Topology for Dynamics

9805584Mischaikow PI建议开发技术，算法和计算机代码，通过这些技术，康利指数将成为一个有效的计算工具。 有各种各样的问题促使这种发展。 主要的问题是，能够计算不变集的非线性动力系统所产生的微分方程或物理实验。 由于一般情况下不变集在小扰动下会发生剧烈的分叉，数值误差、参数漂移和噪声往往会使不变集的直接近似在数值上不稳定，Conley指标作为一个代数拓扑量，对于扰动是非常稳定的。 同时，它足够精细，可以提供有关不变集结构的有用信息（例如符号动力学）。 计算Conley指数包括两个步骤：（1）近似动力学，（2）计算代数拓扑量。 这两个问题都在这个项目中得到解决。具体应用包括与实验学家合作分析和控制混沌系统，以及高维常微分方程和偏微分方程的低维吸引子的严格数值分析。

9805584MischaikowThe PI proposes to develop techniques, algorithms, and computer code through which the Conley index will become an effective computational tool. There are a wide variety of problems motivating this development. The primary issue is that of being able to compute invariant sets for nonlinear dynamical systems arising from differential equations or physical experiments. Since in general invariant sets can undergo dramatic bifurcations under small perturbations, numerical errors, parameter drift, and noise tend to make direct approximations of invariant sets numerically unstable.Being an algebraic topological quantity, the Conley index is extremely stable with respect to perturbations. At the same time it is fine enough to provide useful information (e.g. symbolic dynamics) about the structure of the invariant set. Computing the Conley index involves two steps: (1) approximating the dynamics, and (2) computing algebraic topological quantities. Both of these issues are being addressed in this project. Specific applications include joint work with experimentalists to analyze and control chaotic systems, and the rigorous numerical analysis of low dimensional attractors of high dimensional ordinary differential equations and partial differential equations.