CAREER: Chaotic Dynamics of Systems with Noise

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

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.