Interpretation of Computer Simulations and Experimental Data from Chaotic Processes

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Sauer9971798The investigator studies computational aspects of nonlinear dynamical systems, with emphasis on questions that have multidisciplinary implications in the sciences and engineering. The first main area involves the validity of long-term computer simulations of typical nonhyperbolic chaotic systems. The question is whether long simulation trajectories can be assumed to approximately match true system behavior. This is a far-reaching question that goes to the heart of numerical computation. Numerical analysis is typically concerned with algorithm properties that are local in time, but the possibility of chaotic trajectories in nonlinear systems is causing this view to shift markedly. The project involves the development of quantitative laws for the expected time between breakdowns, or mismatches, between computer simulation trajectories and the correct trajectories of computer models, and investigation of statistical quantities that may be severely miscalculated because of the mismatch. The second major area is ongoing work by the investigator on the interpretation of laboratory experiments that generate aperiodic data streams. Using univariate or multivariate time series or inter-event time intervals produced by a deterministic physical process, the phase space of the process can be faithfully reconstructed as the basis for applications such as system identification, filtering, prediction and control. The mathematical foundations of these applications has been the topic of ongoing research by the investigator and coworkers. The project attempts to widen the areas of validity of these methods, and increase their power for the study of complex systems, especially in the presence of observational noise.Computer simulations are an essential part of modern science. The importance of the correct interpretation of long-term computer simulations of nonlinear models is increasing as modeling replaces experiments and expensive traditional methods of design in areas as diverse as biotechnology, rational drug design, meteorology, satellite orbit trajectory design, wind tunnel testing, and neurophysiology. The questions explored in this project are critical to the fundamental understanding of the results of complex simulations of nonlinear processes. One goal of this project is to explore these questions in physically relevant models, and in particular to isolate and quantify the limitations of these representations, especially for the purpose of long-term modeling. The second focus of this research is the interpretation of data collected from chaotic systems in laboratory experiments and nature. Complex deterministic time series are being explored as key information in physical, chemical, engineering and biological/medical settings. As an example, neuron firing data from hippocampal cells of mammals and other small neural systems are studied by the investigator in conjunction with a group of medical researchers who have recently relocated at George Mason University, with the purpose of detecting deterministic information processing in the brain. The investigator has done previous work on expanding these conceptual foundations and developing related computational implementations, and plans to widen their areas of validity and increase their power for the study of complex systems in natural, experimental, and engineering-related contexts. New techniques for these applications are developed in this project, involving new computation techniques used in conjunction with existing signal processing methods.