Investigating Nonlinear Dynamics with Topological Methods

University Of Delaware, Newark DE

Investigators

Abstract

The project is an investigation of the interplay of topology and nonlinear dynamics from both applied and theoretical viewpoints. Complicated topology arises naturally in nonlinear dynamical systems and frequently "carries" the dynamics of the system. Understanding this topology is essential to understanding the dynamics and the long-term behavior of the system. Planned are (i) further theoretical study of the topology present in "most" (i.e., generic) dynamical systems with a given property, (ii) applied study of stirring and turbulence (with chemical engineers at Rutgers), (iii) theoretical and applied study of area-preserving systems, (iv) an investigation of how topology and invariant measures are related in one-dimesional dynamical systems, and (v) further study of topological horseshoes (with J. A. Yorke and others from the Maryland group). The project builds on work already done by the investigator both alone and in collaboration with other mathematicians and scientists. The tools of both topology and dynamical systems are needed and will be used to carry out the project. Theoretical results concerning stirring will be backed up by actual experiment and observation in the lab. Whenever a nonlinear dynamical system, such as the weather, or the motion of different fluids as they are stirred, or the solar system, is "operating", interesting topology arises as sets are mapped back across themselves. Understanding these sets is crucial to understanding the complicated behavior of nonlinear systems. Sometimes the complicated topology can be observed as it forms, and is a marker for certain kinds of complicated dynamics. Can this be made precise? Can it be "quantified"? Consider stirring, a phenomenon of special interest to the investigator, for example. Stirring occurs in every kitchen, and seems to be a simple process. But this is far from the case: Stirring occurs in industrial settings and in nature in many different contexts as chemicals or medicines, or hot and cold air, or hot and cold water, are mixed. Most often in industry achieving homogeneity of the mixed substance efficiently is the goal. Experimental and simulated results by chemical engineers demonstrate that those processes used today do not result in homogeneity. How can those processes be improved? That is one of the problems to which the results of this project should be applicable. More generally, the project planned is a study of the topology of complicated dynamical systems, and both the implications of the presence of complicated topology for nonlinear systems and vice versa.