Mathematical Sciences: Large-Scale-Ratio Space-Time Chaos

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Greenside The investigator undertakes computational and theoretical efforts to understand extensively chaotic nonequilibrium systems such as those studied experimentally in fluid, laser, and chemical systems by Ahlers, Behringer, Gauthier, Gollub, Kolodner, and Swinney. A spatially-homogeneous chaotic system is defined to be extensive if its fractal dimension grows linearly with the volume of the system. Theorists hope that extensively chaotic systems may be amenable to a thermodynamic-like description that characterizes complex systems through laws relating intensive and extensive macroscopic (coarse-grained) variables. Specific research problems studied are: the identification and analysis of simplified mathematical models that have experimentally relevant features of extensive chaos; a comparison of different spatial correlation lengths for quantifying extensive chaos; a study of the variation of dimension density and other intensive variables with system inhomogeneities; the development and testing of algorithms for estimating fractal dimension densities from time series; the analysis of the lengthscales and timescales involved with the transition between non-extensive and extensive chaos; the identification and testing of Langevin descriptions of extensively chaotic regimes; and the development of related numerical algorithms and optimized computer codes for scalable parallel computers such as the Duke 32-node CM-5 computer. Many nonequilibrium physical systems---both natural and manmade---are complex in that they evolve nonperiodically in both space and time. An increasingly important theme in many modern areas of science and technology is how to quantify this complex behavior. Why do systems become complex as one varies parameters, how does one complex state differ from another, and how does transport of energy and matter depend on complexity? Recent experimental advances in the study of fluid flow, of liquid crystals, of nonlinear optics, and of chemical excitable media have identified a new and important class of complex systems to understand: large homogeneous sustained nonequilibrium systems that undergo a sudden change from a time-independent homogeneous state to a non-transient temporally and spatially disordered state (spatiotemporal chaos). This research uses a combination of mathematical analysis and large-scale parallel computing to explore the properties of carefully chosen models of spatiotemporal chaos. Of particular interest is to understand how complexity grows with the size of a physical system (extensive chaos) and whether a thermodynamic-like description can be found in the limit of infinite system size. An improved understanding of extensive chaos should improve the design, control, and optimization of engineering applications, improve the forecasting of nonperiodic time series, and improve the validation, efficiency, and accuracy of large-scale computer simulations for complex systems.