Dynamical Systems

New York University, New York NY

Investigators

Abstract

Four projects on the mathematical theory of dynamical systems are proposed. Each project contains a cluster of problems with a common theme. The first pertains to strange attractors with strong dissipation and a single direction of instability. Extensions of a general theory developed under a previous grant to arbitrary phase-dimensions are proposed, as are applications to concrete problems such as nonlinear oscillators. The second project concerns the statistical behavior of dynamical systems with predominantly hyperbolic behavior. The focus of the proposed research is on mechanisms leading to various rates of correlation decay in both discrete and continuous times. The topic of the third project is lattice dynamical systems. A systematic analysis of the aggregate behavior of large numbers of dynamical systems coupled together is proposed. The final project proposes dynamical systems methods of solution for three unrelated problems on the Schrodinger operator, kinematic fast dynamo and Navier-Stokes equations. As a branch of mathematics, dynamical systems is concerned with the time evolutions of processes governed by certain underlying laws. A primary goal of the subject is to develop unifying mathematical theories to explain observed phenomena and predict future occurrences. In this proposal, the investigation of a number of models amenable to mathematical analysis and with potential applications to the physical and biological sciences is proposed. It has been known for some time that relatively simple laws can lead to complicated dynamics. The first part of this proposal focuses on systems with chaotic behavior. Two topics are proposed: an analysis of strange attractors and a statistical theory of mixing. (Strange attractors are highly complex objects which capture the long term behaviors of dissipative dynamical systems; they have been observed frequently in nature and in simulations but have thus far resisted rigorous analysis.) Other projects proposed include the relations between aggregate properties of large dynamical systems and those of their individual components, and a few problems from physics and hydrodynamics which the principal investigator believes can be solved by the methods of dynamical systems. Four projects on the mathematical theory of dynamical systems are proposed. Each project contains a cluster of problems with a common theme. The first pertains to strange attractors with strong dissipation and a single direction of instability. Extensions of a general theory developed under a previous grant to arbitrary phase-dimensions are proposed, as are applications to concrete problems such as nonlinear oscillators. The second project concerns the statistical behavior of dynamical systems with predominantly hyperbolic behavior. The focus of the proposed research is on mechanisms leading to various rates of correlation decay in both discrete and continuous times. The topic of the third project is lattice dynamical systems. A systematic analysis of the aggregate behavior of large numbers of dynamical systems coupled together is proposed. The final project proposes dynamical systems methods of solution for three unrelated problems on the Schrodinger operator, kinematic fast dynamo and Navier-Stokes equations. As a branch of mathematics, dynamical systems is concerned with the time evolutions of processes governed by certain underlying laws. A primary goal of the subject is to develop unifying mathematical theories to explain observed phenomena and predict future occurrences. In this proposal, the investigation of a number of models amenable to mathematical analysis and with potential applications to the physical and biological sciences is proposed. It has been known for some time that relatively simple laws can lead to complicated dynamics. The first part of this proposal focuses on systems with chaotic behavior. Two topics are proposed: an analysis of strange attractors and a statistical theory of mixing. (Strange attractors are highly complex objects which capture the long term behaviors of dissipative dynamical systems; they have been observed frequently in nature and in simulations but have thus far resisted rigorous analysis.) Other projects proposed include the relations between aggregate properties of large dynamical systems and those of their individual components, and a few problems from physics and hydrodynamics which the principal investigator believes can be solved by the methods of dynamical systems.