Computational Methods in Applications of Nonlinear Dynamics

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In recent years, modeling of dynamic phenomena in engineering and science, from high-energy lasers to weather prediction, has dramatically increased in accuracy due to more sophisticated treatment of nonlinear effects. The field of nonlinear dynamics, which is focused on exactly how to carry out the improvements, has experienced a parallel increase in importance. However, more accurate modeling leads to more delicate computational simulations. One focus of this project is to investigate issues of validity of computer simulations. We plan to study the interaction between nonlinear dynamics and computer arithmetic, that surprisingly often proves to have a major impact on simulation outcomes.The second focus of the project is research on fundamental questions in the interpretation of experimental multivariate time series and spike train data, including data from physical experiments, epileptic seizures, and signal processing. Complex deterministic time series are being identified in physical, chemical, engineering and biological/medical settings. Techniques will be developed to reconstruct the dynamical behavior of the process, with potential to predict or control the process.The broader impact of the project derives from integration of the research topics in the PI's undergraduate and graduate teaching, the support of graduate student research, and the enhancement of research infrastructure through collaboration with non-mathematician collaborators. There are close connections between the fundamental questions posed in the proposal and introductory numerical analysis and dynamical systems at the undergraduate and graduate levels; the PI has co-authored a text on the latter topic and is finishing an undergraduate text on the former (publication expected Dec. 2005), demonstrating an ongoing commitment to transferring benefits of research into teaching.