Dynamical Systems Approach to Computer Simulation Accuracy and Data Analysis

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Sauer0208092 The investigator studies computational aspects of nonlineardynamical systems, emphasizing the role of chaotic dynamics inthe interpretation of long-time simulation outcomes and data fromcomplicated dynamical processes. The work on simulation is anongoing study of subtle biases occurring in deterministicmodeling that have macroscopic effects on outcomes. It is anopen question whether it is possible in principle for long-termcomputer simulations of typical nonhyperbolic chaotic systems toapproximately match true system behavior. The second major areais ongoing work on the interpretation of physical and biologicalexperiments that generate aperiodic data. There is a long historyof applying random modeling to time series and spike timing datacollected from laaboratory and natural processes. Theinvestigator syudies ways to use dynamical systems techniques,including embedding theory and spike train reconstruction, toextend the power of deterministic modeling for experimental data. Computer simulation is a critical ingredient of modernscience. It is important to know whether solutions of amathematical model used for simulation can be expected to beaccurate qualitative and quantitative representations of thenatural phenomena being modeled, in the face of small modelingerrors, and whether computer solution of the model can representsolutions of the model, in the face of small floating pointerrors. This reseach builds a foundation for answering questionslike this for physically relevant models. In particular theinvestigator is involved in isolating and quantifying thelimitations of deterministic representations, especially for thepurpose of long-term modeling. The second major focus of theproject is the interpretation of data collected from experimentalsystems and nature when no first-principles mathematical orcomputer model is available. If time traces of physicallyrelevant quantities can be measured from the process, techniquesexist to attempt to reconstruct the dynamical behavior of theprocess, with potential to predict or control the process.Complex deterministic time series are being identified inphysical, chemical, engineering and biological/medical settings.The investigator has made previous progress on expanding theseconceptual foundations and developing related computationalimplementations, and is working to increase their power for thestudy of complex systems in scientific and engineering-relatedcontexts.