Computational Analysis of Large Dynamical Systems

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。这项工作包括三个与多自由度动力系统的计算分析相关的项目。 第一个涉及耦合的生物振荡器和可兴奋的元件由波动的外部刺激驱动的网络。 其目标是阐明，在一类生物相关的架构，网络的结构和再现性，或可靠性，其响应之间的关系。 第二个项目涉及宏观输运过程的出现时，一个开放的系统耦合到不平等的热源在其边界。 这个项目的重点是一类原型模型，包括确定性和随机微观动力学。 这里的目的是深入了解在一个具体的一类模型系统的非平衡稳态的属性。 第三个项目旨在开发计算统计平均值的有效数值算法，例如，李雅普诺夫指数，经常用来描述非线性动力系统。 待开发的算法基于利用待计算的量的近似先验知识，例如，系统不变测度的近似知识。 这将通过耦合所讨论的系统的多个模拟来完成，并使用这种耦合来产生具有潜在的显著较小方差的无偏估计。 算法在各种生物和物理环境中的有效性，包括上述项目，将被调查。具有许多强非线性自由度的动力系统出现在许多科学和技术问题中。 它们的分析和模拟往往是困难的，因为它们的相互作用的复杂性和它们往往混乱的动态。 包括这项工作的项目试图了解这样的大型动力系统在一些特定的设置，并开发通用的，有效的数值算法计算非线性动力系统的相关统计特性。 这项研究的预期结果可能会导致对一系列现象的更深入的了解，包括生物神经网络编码信息的能力以及具有复杂微观相互作用的空间扩展系统中宏观能量和物质传输的出现。 待开发的算法可能适用于其他应用领域，例如，随机化学动力学 预计这些项目将导致跨学科合作，例如，与生物科学家，并为研究生培训的机会。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This work consists of three projects related to the computational analysis of dynamical systems with many degrees of freedom. The first concerns networks of coupled biological oscillators and excitable elements driven by fluctuating external stimuli. The goal is to elucidate, within a class of biologically-relevant architectures, the relation between a network's structure and the reproducibility, or reliability, of its response. The second project concerns the emergence of macroscopic transport processes when an open system is coupled to unequal heat reservoirs at its boundaries. This project focuses on a prototypical class of models that includes both deterministic and stochastic microscopic dynamics. The aim here is to gain insights into the properties of nonequilibrium steady states in a concrete class of model systems. The third project aims to develop efficient numerical algorithms for computing statistical averages, e.g., Lyapunov exponents, that are frequently used to characterize nonlinear dynamical systems. The algorithms to be developed are based on exploiting approximate prior knowledge of the quantity to be computed, e.g., approximate knowledge of the system's invariant measure. This will be done via coupling multiple simulations of the system in question, and to use such couplings to produce unbiased estimators with potentially significantly smaller variance. The efficacy of the algorithms in various biological and physical settings, including the projects outlined above, will be investigated.Dynamical systems with many strongly nonlinear degrees of freedom arise in many scientific and technological problems. Their analysis and simulation is often difficult because of the complexity of their interactions and their often chaotic dynamics. The projects comprising this work seek to understand such large dynamical systems in some specific settings, and to develop general, efficient numerical algorithms for computing relevant statistical properties of nonlinear dynamical systems. The expected outcome of this research may lead to deeper insights into a range of phenomena, including the ability of biological neural networks to encode information and the emergence of macroscopic energy and matter transport in spatially-extended systems with complex microscopic interactions. The algorithms to be developed are potentially applicable to other application domains, e.g., stochastic chemical kinetics. It is expected that the projects will lead to interdisciplinary collaborations, e.g., with biological scientists, and to opportunities for graduate student training.