Mathematical Sciences: Interpretation of Data from Chaotic Processes

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9305659 Sauer The focus of this research is the interpretation of data collected from chaotic processes and simulations. Using univariate or multivariate time series produced by a deterministic physical process, the phase space of the process can be faithfully reconstructed. 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. Among the many applications of these mathematical foundations are computational techniques for noise reduction (filtering) and time series prediction, to enhance and/or replace conventional methods from the fields of signal processing and time series analysis, which are of limited applicability for data from chaotic processes. A related task that is targeted by this research is the development of reliable diagnostics with which to distinguish linear and nonlinear noise processes from nonlinear dynamics, on the basis of data measured from experiment or the environment. New techniques for these applications are developed in this project, involving methods from numerical analysis used in conjunction with existing signal processing methods. The study of the validity of computer simulations of chaotic processes by means of shadowing is also pursued in conjunction with collaborators. The central goal of this research is to identify and analyze nonlinear behaviors in systems, by measurement and analysis of quantitative signals produced by that system. New techniques allow exploitable deterministic descriptions, using the paradigms of deterministic chaos, for some systems written off earlier as noise, or as too difficult to study. Complex phenomena of this type are being identified in physical, chemical, engineering and biological/medical settings. As an example, hippocampal slices from mammals and other small neural systems are being studied by the investigator, in conjunction with a g roup of medical researchers headed by a neurosurgeon specializing in epilepsy, with the purpose of detecting deterministic information processing in the brain. The first mathematical foundations of the reconstruction techniques pursued in this proposal were disseminated in 1980, which led almost immediately to intensive research across several scientific disciplines, the impact of which has not yet been fully felt. The investigator has done previous work on expanding these conceptual foundations and developing related computational implementations, and under this proposal will widen their areas of validity and increase their power for the study of complex systems in natural, experimental, and engineering-related contexts.