This project focuses on extending and optimizing existing techniques for computational studies of dynamical systems. Recognizing the need for discretizations and truncations of dynamical systems as a first step towards making a numerical study, the investigator and her collaborators focus on techniques that can overcome the corresponding loss of information. In particular, in constructing the finite representation of the system used for numerical computations, they incorporate explicit bounds for discretization, truncation, and other errors accumulated during this process. The finite representation is the dynamical system viewed at a fixed resolution -- information about the precise location of individual points and their trajectories is lost, but coarse, topological structures remain. The investigator will incorporate topological tools, such as algebraic topology and Conley index theory, in computations on the finite representation to detect and prove the existence of dynamics for the original system.
Dynamical systems models are being used throughout society. Some examples include weather models used for hurricane prediction and population models used to study environmental effects on population size and persistence. Currently, many researchers study dynamical systems like these using high powered computer simulations and statistical techniques. On the other end of the spectrum, mathematicians have been able to decipher highly complicated dynamics in more abstract mathematical models. The work described in this proposal aims to serve as a bridge between these two approaches. More specifically, the investigator and her collaborators focus on the development of computational techniques that use sophisticated mathematical tools and yield mathematically rigorous results. The mathematical tools come from the fields of algebraic topology, analysis, numerical analysis, and dynamical systems theory and may be used to decipher some of the phenomena of interest in the studied systems. Prior progress in studying complicated dynamics in models from population ecology and heat convection motivates these continued studies.
