This project combines dynamical systems theory and computational topology in constructing and utilizing rigorous numerical techniques for the study of dynamics in model systems. The primary focus is on constructing, extending and optimizing rigorous numerical techniques and investigating additional settings where computational topology may be used effectively to measure dynamical systems. When possible, the computational results are verified rigorously, typically through the incorporation of error bounds and the use of topological tools. The proposed projects include extending existing techniques for computer-assisted studies and proofs for discrete time dynamical systems and forcing theorems for braided stationary solutions. Two more sets of projects are proposed in areas where the desired rigorous results are still out of reach, although some initial progress has been made. These include measuring parameter sets for systems in a specified family which exhibit chaotic behavior, and the study of spatial and spatio-temporal pattern formation and evolution using computational homology. There are many open research projects in these areas (both shorter and longer term) including the study of some very interesting and fundamental modeling questions concerning appropriate spatial and other scales in models. This grant will fund a teacher-researcher postdoctoral position, undergraduate research projects, and support teaching and research group activities.
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. In addition to continuing work with existing collaborators, the investigator will mentor a postdoctoral teacher-researcher and advise undergraduate research projects in these areas.
