Computational Dynamics and Computer-Assisted Proofs

Dynamical systems present many fundamental challenges to computational study.  These range from the need to adapt finite computations (using a computer) to the task of tracking an infinite number of trajectories, to sensitive dependence, one of the defining properties of chaos, resulting in rapid accumulations of error in tracking even a singe trajectory.  Techniques based on Conley index theory and other tools from topology, dynamics, and graph theory form a framework for overcoming many of these difficulties and, along the way, constructing computer-assisted proofs of dynamics ranging from relatively simple fixed points to chaotic attractors.  Students may focus on studying particular models or work on extending techniques.  One possible project involves using time series data for model construction.  Past projects have included studying population abundance models (for black bears, white-footed mice, and invasive species) as well as algorithm development and extensions of the computational theory.