Characterization of Spatiotemporal Chaos

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9722814 Henry Greenside "CHARACTERIZATION OF SPATIOTEMPORAL CHAOS" Renewal of support is requested for interdisciplinary research (nonequilibrium physics, nonlinear dynamics, and computational mathematics) concerning the characterization of spatiotemporal chaos in large nonequilibrium systems such as those found in biological, chemical, fluid, and laser systems. The research will use numerical methods, parallel computers, and carefully chosen mathematical models to explore two questions: (1) how is spatiotemporal chaos related to the mathematical structure of attractors in phase space? and, (2), can the infinite hierarchy of unstable periodic orbits associated with a strange attractor be used to characterize and control high-dimensional chaos? The proposed research will: (a) yield new methods for analyzing spatiotemporal data accumulated by experimentalists and by computational scientists; (b) provide new information about unstable periodic orbits (UPOs) associated with extensive chaos and suggest new strategies for their stabilization; (c) provide new insights about parallel numerical algorithms for finding unstable periodic orbits of partial differential equations in large spatial domains ; and (d) give valuable training to students and postdocs at Duke in the numerical analysis, simulation, and theory of complex nonlinear dynamical systems. Many problems in basic and applied science require a deeper understanding of sustained nonequilibrium systems in which complicated time-dependent behavior is caused by a steady flux of energy and matter into a system. Examples can be found in almost all areas of science but might include fluid turbulence, fibrillation in a heart, epilepsy in a brain, the design of high-power lasers, and instabilities of electrical power grids. A fundamental question of high current interest for many areas of science is how to re late the spatial structure of a complex nonequilibrium medium to its temporal state, especially when the system varies nonperiodically (chaotically) in time. The proposed research suggests two ways of addressing this question that have yet been incompletely explored by previous research. This research will use applied mathematics and computer simulation to study simplified mathematical models in which spatial and temporal structure can be examined in detail, and then various insights will be tested with experimental data. Eventually, this research should improve engineering control and design of complex systems as well as provide a foundation for developing a fundamental theory of sustained nonequilibrium systems.