摘要：

In this talk, we will present explicit conditions on the Birkhoff-Gustvason normal form of a two-degree-of-freedom Hamiltonian system near an equilibrium point, under which there exist infinitely many periodic orbits on every sphere-like component of the energy surface near the equilibrium point. The equilibrium is supposed to be a nondegenerate minimum of the Hamiltonian. Every sphere-like component of the energy surface that sufficiently close to the equilibrium will contains at least two periodic orbits forming a Hopf link. The method in this study is to check a certain non-resonance condition (established by Hryniewitz, Momin and Salomao2015) for the Hopf link, and then infinitely many periodic orbits follow. This method does not need any global surface of section, as in the study of Hofer, Wysocki and Zehnder 1998. Finally, we apply our results to the spatial isosceles three-body problem, Hill's lunar problem and the Henon-Heiles problem. This is joint work with C. Grotta-Ragazzo and P. A. S. Salomao.