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

果岭边 研究人员进行计算和理论上的努力，以了解广泛的混沌非平衡系统，如实验研究的流体，激光和化学系统的Ahlers，Behringer，Gauthier，Gollub，Kolodner和Swinney。 定义了一个空间均匀混沌系统，如果它的分形维数与系统体积成线性增长，则该系统是广延的。 理论家希望广泛的混沌系统可以服从于一个类似数学的描述，通过与密集和广泛的宏观（粗粒度）变量相关的定律来描述复杂系统。 研究的具体问题有：具有广泛混沌实验相关特征的简化数学模型的识别和分析，量化广泛混沌的不同空间相关长度的比较，维数密度和其他强度变量随系统不均匀性的变化，系统的非均匀性和非均匀性对混沌的影响，混沌的非均匀性对混沌的影响，混沌的非均匀性对混沌的影响，混沌的非均匀性对混沌的非均匀性对混沌的非均匀性的影响，混沌的非均匀性对混沌的非均匀性的影响，混沌的非均匀性对混沌的非均匀性对混沌的非均匀性的影响，混沌的非均匀性对混沌的非均匀性的影响，混沌的非均匀性对混沌的非均匀性对混沌的非均匀性的影响，混沌的非均匀性对混沌的非均匀性对混沌的非均匀性的影响，混沌的非均匀性对混沌的非均匀性对混沌的非均匀性的影响，混沌的非均匀性对混沌的非均匀性对混沌的非均匀性的从时间序列中估计分形维数密度的算法的开发和测试;非扩展混沌和扩展混沌之间过渡所涉及的长度尺度和时间尺度的分析;识别和测试广泛的混沌状态的朗之万描述;和相关的数值算法和优化的计算机代码的可扩展的并行计算机，如杜克32节点CM-5计算机的发展。 许多非平衡物理系统-自然的和人造的-是复杂的，因为它们在空间和时间上都是非周期性地演化。 在许多现代科学和技术领域，一个越来越重要的主题是如何量化这种复杂的行为。 为什么系统会随着参数的变化而变得复杂，一个复杂的状态与另一个复杂的状态有何不同，能量和物质的传输如何依赖于复杂性？ 最近在流体流动、液晶、非线性光学和化学可激发介质的研究中取得的实验进展已经确定了一类新的重要的复杂系统：大型均匀持续非平衡系统，它经历了从与时间无关的均匀状态到非瞬态时间和空间无序状态（时空混沌）的突然变化。 本研究结合数学分析和大规模并行计算，探索精心挑选的时空混沌模型的特性。 特别感兴趣的是了解复杂性如何随着物理系统的大小（广泛的混沌）而增长，以及是否可以在无限系统大小的极限中找到类似于几何的描述。 对广泛混沌的更好理解应该改善工程应用的设计，控制和优化，改善非周期时间序列的预测，并提高复杂系统的大规模计算机模拟的有效性，效率和准确性。

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.