Zhi-Quiang Wang: A Twisting Condition, Resonances, and Periodic Solutions of Hamiltonian Systems
2008 HMC Mathematics Conference on Nonlinear Functional Analysis: Abstracts
Zhi-Qiang Wang (Utah State University)
In this talk, we first discuss a bit of background from the Poincaré-Birkhoff theorem to its applications of finding periodic solutions of Hamiltonian systems. The Poincaré-Birkhoff theorem assures the existence of two fixed points for any area preserving homeomorphisms F on a two-dimensional annulus that satisfy a boundary twist condition. This boundary twist condition states that F advances points on the outer edge of the annulus positively and points on the inner edge negatively. Then we discuss analogues of the twisting condition of the Poincaré-Birkhoff theorem in the setting of higher dimensional Hamiltonian systems, which are related to resonances of the Hamiltonian functions near zero and infinity. Finally we present some recent work on the existence of periodic solutions of Hamiltonian systems under a twisting condition which resembles more in spirit of the classical one in the Poincaré-Birkhoff theorem. This talk is intended for undergraduate students.
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