Abstract:
In this talk, I will speak about the Poincaré--Birkhoff theorem, also known as the Poincaré's last geometric theorem, which was conjectured by Poincaré in 1912. The complete proof was then provided by Birkhoff in 1925. Poincaré and Birkhoff were both motivated by its applications to the search of periodic solutions of conservative dynamical systems. I will discuss some applications of this theorem, especially in the restricted 3-body problem. Later, I will present some higher dimensional generalization of this result for Hamiltonian flows. At the end, I will speak about our recent work in this direction, where we explored the existence of multiple periodic solutions for coupled Hamiltonian systems with distinct structural behaviors. One system exhibits the Poincaré--Birkhoff twist-type structure, while the other is characterized by a singular nonlinearity near the origin.
參考連結
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