9626197 Sauer This research focuses on computational aspects of chaotic dynamics, with emphasis on questions that have implications across the sciences and engineering. The first main area involves the question of whether it is possible, even in principle, for long-term computer simulations of typical nonhyperbolic chaotic systems to approximately match true system behavior. Part of this research involves developing quantitative laws of the expected waiting time between breakdowns, or mismatches, between computer simulation trajectories and the trajectories of computer models. The second major area is ongoing work by the investigator on the interpretation of physical and biological experiments that generate aperiodic time series. 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. Topological and diffeomorphic embedding theorems guaranteeing this in various physically verifiable contexts have been the results of previous and ongoing research by the investigator and coworkers. Computer simulations are a staple of modern science. National policy decisions are being made which rely partially on the interpretation of long-term computer simulations of nonlinear models. The questions explored in the proposal are critical to the analysis of simulations of nonlinear processes. One goal of this proposal 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 experiments and nature. Complex deterministic time series are being identified in physical, chemical, engineering and biological/medical setti ngs. As an example, hippocampal slices from mammals and other small neural systems are studied by the investigator, in conjunction with a group of medical researchers headed by a neurosurgeon specializing in epilepsy, 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 works 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 methods from numerical analysis used in conjunction with existing signal processing methods.
