Vector Fields, Angular One-Forms and Periodic Orbits

Chris Byrnes E. H. and F. G. Skinner Professor of Systems Science and Mathematics Washington University St. Louis, MO, USA Date: February 28, 2008 Location: LSE 104 Time: 4:00-5:00 PM Refreshments: PSA 206 at 3:15 PM

Abstract

Periodic phenomena are pervasive in nature and in engineered systems. They are exhibited, for example, in idealized models of the solar system and in observed circadian rhythms by which basic biological functions are believed to be regulated.The GPS system has 27 satellites rotating about the earth in precise stable orbits, each orbit being provided in an almanac in every GPS receiver. As another class of examples, electronic devices producing stable periodic signals underlie the electrification of the world and wireless communications.

We will begin this talk with an historical review of classica existence ciriteria for periodic orbits on a smooth surface, with or without boundary. As an example, we will analyze a stable oscillating circuit that is in widespread commercial use today. In fact, every cell-phone has two, one used for transmitting 0's and the other for transmitting 1's at stable radio frequencies. In this example, the vector field has an ``angular" one-form, a concept with roots in earlier work of G. D. Birkhoff. Roughly speaking, an angular one-form is a closed nonsingular one-form which is a generalized form of angular velocity analogous to the interpretation of a Lyapunov function as a generalized form of energy.

Forty years ago Smale asked whether every nonvanishing smooth vector field on the solid torus had a periodic orbit. In 1996, G. and K. Kuperberg answered this in the negative. Nonetheless, in this talk we present a series of positive results on the existence of periodic orbits for any vector field X on n-dimensional solid tori having an angular one-form . Moreover, using the validity of the Poincare Conjecture in all dimensions we prove a converse theorem in the spirit of Lyapunov theory: If a vector field X on R^n. has an asymptotically stable periodic orbit, then there exists a neighborhood M of the orbit which is homeomorphic to a solid torus, on which X has an angular one-form. .

These are corollaries of a Main Theorem, which is valid for a broad class of n-dimensional compact manifolds (with or without boundary). In closing, we illustrate the Main Theorem in the case of 3-dimensional manifolds, using Thurston’s Geometrization Program.