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 that regulate basic biological functions. Electronic devices producing stable periodic signals underlie the electrification of the world and wireless communications. In this talk, we present conditions that guarantee that a periodic motion exists for a dynamical system. We will begin by analyzing a stable oscillating circuit that is in widespread commercial use in electronic communications, e.g., in every cell phone. This simple example shares some important features with a nonlinear three-dimensional model of an AC motor that uses oscillations in its magnetic field to produce stable mechanical rotations. These features can be expressed very simply in terms of the vector field defining the dynamics and an "angular variable", a concept with roots in earlier work of G. D. Birkhoff. Not coincidentally, both of these systems evolve on a solid torus (in two and three dimensions, respectively). Forty years ago, Smale asked whether every nonvanishing smooth vector field X on the solid torus had a periodic orbit. In 1996, G. and K. Kuperberg answered this in the negative. Nonetheless, under the conditions we identified, periodic orbits exist for any vector field on an n-dimensional solid torus. Indeed, the language of fields and forms allows one to use global topological methods, for example the fruitful combination of homotopy and cobordism, to understand the existence of periodic phenomena. Moreover, using these methods in higher dimensions and the proofs of Poincare Conjecture in dimensions three and four, we prove the existence of an invariant solid torus and an angular variable are necessary for the existence of an asymptotically stable periodic orbit. These results 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.
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Prof. Christopher Byrnes is the Edward H. and Florence G. Skinner Professor of Systems Science and Mathematics at Washington University in St. Louis. He received an Honorary Doctorate of Technology from the Royal Institute of Technology (KTH) in Stockholm in 1998 and was elected a Foreign Member of the Royal Swedish Academy of Engineering Sciences in 2002. Chris Byrnes is author of more than 250 technical papers and books. With his coauthors, he has (twice) received the IEEE George Axelby Award for best paper in the IEEE Transactions in Automatic Control and an IFAC Best Paper Award. Moreover, he was awarded the SIAM Reid Prize in 2005 and the IEEE Hendrik W. Bode Prize in 2008. In the Summer Semester 2009, he has the Giovanni Prodi Chair in Nonlinear Analysis at the University of Wuerzburg. The next academic year, Chris Byrnes will be Guest Professor at KTH in Stockholm.
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