Dynamical Systems Approach to Computer Simulation Accuracy and Data Analysis

George Mason University, Fairfax VA

Investigators

Abstract

Sauer 0208092 The investigator studies computational aspects of nonlinear dynamical systems, emphasizing the role of chaotic dynamics in the interpretation of long-time simulation outcomes and data from complicated dynamical processes. The work on simulation is an ongoing study of subtle biases occurring in deterministic modeling that have macroscopic effects on outcomes. It is an open question whether it is possible in principle for long-term computer simulations of typical nonhyperbolic chaotic systems to approximately match true system behavior. The second major area is ongoing work on the interpretation of physical and biological experiments that generate aperiodic data. There is a long history of applying random modeling to time series and spike timing data collected from laaboratory and natural processes. The investigator syudies ways to use dynamical systems techniques, including embedding theory and spike train reconstruction, to extend the power of deterministic modeling for experimental data. Computer simulation is a critical ingredient of modern science. It is important to know whether solutions of a mathematical model used for simulation can be expected to be accurate qualitative and quantitative representations of the natural phenomena being modeled, in the face of small modeling errors, and whether computer solution of the model can represent solutions of the model, in the face of small floating point errors. This reseach builds a foundation for answering questions like this for physically relevant models. In particular the investigator is involved in isolating and quantifying the limitations of deterministic representations, especially for the purpose of long-term modeling. The second major focus of the project is the interpretation of data collected from experimental systems and nature when no first-principles mathematical or computer model is available. If time traces of physically relevant quantities can be measured from the process, techniques exist to attempt to reconstruct the dynamical behavior of the process, with potential to predict or control the process. Complex deterministic time series are being identified in physical, chemical, engineering and biological/medical settings. The investigator has made previous progress on expanding these conceptual foundations and developing related computational implementations, and is working to increase their power for the study of complex systems in scientific and engineering-related contexts.