Dynamics, Chaos, and the Conley Index
Discrete-time dynamical systems modeled by iteration of continuous maps exhibit a wide variety of interesting behaviors. One illustrative example is the one-dimensional logistic model. For the logistic model, chaotic dynamics may be proven via a topological conjugacy onto an appropriate subshift of finite type, a symbolic system for which a proof of chaos is attainable. Analysis and proofs of dynamics for other discrete-time models, especially in dimensions larger than one, often prove to be more challenging. In this talk, I will present methods for constructing and analyzing outer approximations, finite representations of discrete-time models that are amenable to computational studies and computer-assisted proofs. These methods rely heavily on Conley index theory, an algebraic topological generalization of Morse Theory. Sample results for the two-dimensional Henon map and the infinite-dimensional Kot-Schaffer model from ecology will serve as illustrations of the methods.
Refreshments in Math Commons Room at 3:30pm