Topic description: Errors, whether from computer round-off or other effects, are unavoidable in numerical studies of dynamical systems. For interesting, especially chaotic systems, these errors may accumulate in time, obscuring the true behavior of the system. Recently, techniques based on ideas from topology have been adapted to incorporate bounded error in numerical computations. For example, these techniques applied to the Kot-Schaffer and Henon models resulted in the location (up to machine precision) and rigorous verification of the existence of (unstable) periodic orbits, connecting orbits, and more complicated (chaotic) invariant sets.
Research opportunities: Existing code and software offers a natural starting point for student research, and there are still many interesting questions to be addressed on both the theoretical and practical fronts. For example, questions about the optimality of existing algorithms and extensions of the current approach were the focus of an (undergraduate) REU project at Cornell University in summer 2006. This area contains a wide variety of projects for undergraduate research incorporating, among other fields, dynamical systems, graph theory, linear algebra, and topology.
Suggested prerequisites: Math 302, Math 426, experience with Matlab, C++
Contact: Sarah Day