Colloquia Patavina: Vector Fields, Angular One-Forms and Periodic Orbits


Martedi' 2 Ottobre 2007 alle ore 16:00 in aula 1A/150 della Torre Archimede il Professor Christopher I. Byrnes della Washington University terra' la decima conferenza della serie Colloquia Patavina dal titolo "Vector Fields, Angular One-Forms and Periodic Orbits".

La Commissione Colloquia
M. A. Garuti, M. Pavon, M. Pitteri, F. Rossi

Vector Fields, Angular One-Forms and Periodic Orbits

Christopher I. Byrnes

Martedi' 2 Ottobre 2007, ore 16:00
Aula: 1A/150, Torre Archimede

- Riassunto
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 rythyms by which basic biological functions are believed to be regulated. Electronic devices producing stable periodic signals underly the electrification of the world and
wireless communications. For example, every cell-phone has two antennae which transmit at separate, stable radio frequencies, one used for transmitting 0's and the other for 1's. As another example, the GPS system has 27 satellites rotating about the earth in stable orbits, each orbit being provided in an almanac in every GPS receiver. We are interested in the basic question as to whether a nowhere zero vector field on a smooth compact manifold, with or without boundary, has a periodic orbit. For example, forty years ago Smale asked whether every nonvanishing smooth vector field on the solid torus had a periodic orbit. In 1995, K. Kuperberg answered this in the negative. Nonetheless, in this talk we present a series of positive results on the existence of periodic orbits that are valid, under some fairly natural hypotheses, in much greater generality than the setting of Smale's original question. In particular, we present results on the existence of periodic
orbits for vector fields having an "angular" one-form. 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. We illustrate these results in the case of 3-dimensional manifolds
and for n-dimensional solid tori, using Thurston's Geometrization Program and, in particular, the solution of the Poincare conjecture

-Breve curriculum
Christopher Byrnes received his bachelors degree in mathematics from Manhattan College in 1971, and his masters and doctorate in mathematics from the University of Massachusetts in 1973 and 1975.
>From 1975 to 1978, he was an instructor at the University of Utah.
>From 1978-1984, he was an assistant professor, then an associate professor in the Division of Applied Science at Harvard University.
>From 1984-1989, he was research professor of engineering and mathematics at Arizona State University.
In 1989, he came to Washington University as chairman and professor in the Department of Systems Science and Mathematics. In 1991, he became dean of the School of Engineering and Applied Science. He has held visiting
appointments at various institutions including: Kungliska Tekniska Hogskolan, Stockholm, Sweden; Ceremade, Universite Paris - Dauphine, Paris, France; Universita di Roma, Rome, Italy; Soviet Academy of Sciences, Irkutsk, Kiev Moscow, USSR; IIASA, Laxenburg, Austria; University of Tokyo, Tokyo, Japan; Universiteit Groningen, The Netherlands, and Universitat Bremen, Bremen, BRD.
During the last decade impressive progress has been made in the understanding of nonlinear dynamical systems, using newly developed geometric, algebraic, and analytic methods. Nonlinear distributed parameter systems, in particular, provide models for many important physical systems, including rigid spacecraft with feasible appendages, lightweight flexible structures (such as elastic robot arms), and air flow across an airport.
One of the long range goals of Dean Christopher Byrnes research is the development of a systematic methodology, similar in intuitive appeal and scope to classical automatic control, for designing feedback laws capable of stabilizing or perhaps even shaping the time response of nonlinear lumped and distributed parameter systems.
To this end, he and his students have been studying two closely allied approaches, which have their origins in two aspects of the control of complex dynamical systems: the development and analysis of a system description as accurate as possible and the alternate development of reduced order models, capturing some of the dominant dynamical effects.
Dr. Byrnes is also active in research programs in digital signal processing and speed processing.

-Awards and Honors
-Honorary Doctor of Technology, Royal Institute of Technology, Stockholm, Sweden, 1998
-Fellow of the Academy of Science of St. Louis, 1998
-IFAC Best Paper Award for three-year period 1990-1992, awarded at the IFAC World Congress in Sydney, Australia, 1993
-George Axelby Prize for Best Paper in IEEE Transactions on Automatic Control, 1990
-IEEE Fellow, 1989
-The Graduate College Distinguished Research Award, Arizona State University, 1988
-JSPS Fellow (Japan Society for the Promotion of Science), 1986
-Case Centennial Scholar, Case Western Reserve University, 1980
-Pi Mu Epsilon, Sigma Xi, and Tau Beta Pi
-Output Regulation of Nonlinear Systems, IEEE Transactions on Automatic Control, vol. 35, No. 2, February 1990, Winner of the George Axelby Prize for Best Paper in IEEE Transactions on Automatic Control, 1990 (with A. Isidori).
-A Complete Parameterization of all Positive Rational Extensions of a Covariance Sequence, IEEE Transactions on Automatic Control, vol 40, 1995, 1841-1857 (with A. Lindquist, S.V. Gusev, and A.V. Natveev).

Rif. int. F. Rossi

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