The 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.
