CAREER: Chaotic Dynamics of Systems with Noise

动力系统是随时间演化的实验配置。这个通用的框架包括自然科学中的大量模型，例如，生态系统中动物和植物种群的动态，以及过往船只后面湍流尾流的形状和形式。该提案中的工作涉及动力系统的混沌行为，例如，通过对初始条件的敏感依赖-一种也被称为“蝴蝶效应”的现象，其中对实验初始准备的微小变化会导致随着时间的推移而产生截然不同的结果。 混沌在具有实际意义的系统中几乎无处不在。 日常生活中的一个例子是天气预测：例如，敏感的相关性解释了为什么飓风的预测路径会随着未来的预测而变宽，因为我们对当前天气系统的测量中的微小不精确会随着时间的推移而被“放大”。这项职业资助将PI的研究工作整合到教育计划中，包括本科生的定向阅读计划和研究机会;介绍新的本科生和研究生课程，使用概率论工具处理混沌动力系统;以及数学研究生暑期学校，帮助他们进入这个引人注目和快速发展的领域的研究前沿。 从低维玩具模型到高维系统，混沌几乎是一个普遍存在的特征，包括各种各样的由演化PDE所禁止的无限维动力学，如控制粘性流体运动的不可压缩Navier-Stokes方程。 平滑遍历理论的工具提供了一个抽象的框架，通过研究它们的统计特性，如相关性的衰减，来理解动力系统中的混沌行为。 然而，它仍然是一个重大的挑战，将这些抽象的工具集合应用到许多具有实际意义的系统中。 例如，尽管有相当多的支持数值证据，它仍然是一个开放的问题，以证明奇里科夫标准映射有一个积极的李雅普诺夫指数的正体积子集相空间。 PI和其他人最近的工作已经发现，少量的非退化噪声被相空间中存在的拉伸“放大”，从而使确定动力学是否是混沌的问题变得容易处理。 PI将这一原理应用于与实际应用相关的各种动力系统，例如流体动力学。 例如，PI和合作者表明，不可压缩流体中的被动示踪剂（拉格朗日）流动--例如悬浮在流体中的尘埃所描绘的运动--在上述意义上是混乱的。这导致了巴彻勒定律的数学证明，定量预测溶质浓度分布中小尺度结构的形成，例如倒入一杯咖啡的牛奶滴形成的滚滚图案。 本项目旨在通过将平滑遍历理论应用于更复杂的动力系统，例如，不可压缩速度场在噪声存在下的Navier-Stokes方程的演化（所谓的流体欧拉动力学）。沿着这些路线的一个长期目标沿着是一个详细的，数学上严格的调查，从层流和有序的行为过渡到混乱和不可预测的动力学在许多流体系统中看到的雷诺数的增加。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Dynamical systems are experimental configurations that evolve in time. This versatile framework includes a vast array of models in the natural sciences, for example, the dynamics of populations of animals and plants in an ecological system, and the shape and form of a turbulent wake behind a passing ship. The work in this proposal concerns chaotic behavior of dynamical systems, signified, for instance, by sensitive dependence on initial conditions – a phenomenon also known as “the butterfly effect,” in which tiny changes to the initial preparation of an experiment lead to drastically different outcomes as time progresses. Chaos is nearly ubiquitous in systems of practical interest. An example from everyday life is weather prediction: sensitive dependence explains, for instance, why the predicted path of a hurricane widens as one forecasts further into the future, as small imprecisions in our measurements of the present weather system are “amplified” as time progresses. This CAREER grant will integrate the PI’s research work into educational programs, including directed reading programs and research opportunities for undergraduate students; the introduction of new undergraduate and graduate courses on the treatment of chaotic dynamical systems using tools from probability theory; and a summer school for graduate students in mathematics helping to bring them to the research front of this compelling and rapidly evolving field. Chaos is a nearly ubiquitous feature of dynamical systems of practical interest, from low-dimensional toy models to high-dimensional systems, including a wide variety of infinite-dimensional dynamics proscribed by evolutionary PDE such as the incompressible Navier-Stokes equations governing the motion of a viscous fluid. The tools of smooth ergodic theory provide an abstract framework for understanding chaotic behavior in dynamical systems through the study of their statistical properties, such as the decay of correlations. However, it remains a major challenge to apply this collection of abstract tools to many systems of practical interest. For instance, despite considerable supporting numerical evidence, it remains an open problem to prove that the Chirikov standard map has a positive Lyapunov exponent on a positive-volume subset of phase space. Recent work of the PI and others has uncovered that a small amount of nondegenerate noise is “amplified” by the presence of stretching in phase space, rendering tractable the problem of determining whether the dynamics is chaotic. This principle has been applied by the PI to a variety of dynamical systems relevant to practical applications, such as fluid dynamics. For instance, it was shown by the PI and collaborators that passive tracer (Lagrangian) flow in an incompressible fluid — such as the motion traced out by a mote of dust suspended in a fluid — is chaotic in the above sense. This led to a mathematical proof of Batchelor’s law, a quantitative prediction of the formation of small-scale structures in the concentration profile of a solute, such as the billowing patterns formed by drops of milk poured into a cup of coffee. This project seeks to build off these successes by applying smooth ergodic theory to more complicated dynamical systems, e.g., the evolution of an incompressible velocity field by the Navier-Stokes equations in the presence of noise (the so-called Eulerian dynamics of a fluid). A long-term goal along these lines is a detailed, mathematically rigorous investigation into the transition from laminar and ordered behavior to chaotic and unpredictable dynamics seen in many fluid systems as Reynolds’ number is increased.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.