Resumen de Critical Periods of the Sum of Two Quasi-Homogeneous Hamiltonian Vector Fields

Ziwei Zhuang, Changjian Liu

• In this paper, we consider the period annulus of a family of Hamiltonian systems with a center at the origin. The Hamiltonian of this family is the sum of two quasihomogeneous polynomials with the same weights but different quasi-degrees. We prove that the period annulus of the origin has at most one simple critical period under no restrictions on the weights, which extends a previous result. In particular, we give a positive answer to the 12th Gasull’s problem proposed in his paper (SeMA J 78(3):

  233–269, 2021). Moreover, the unique critical period is reachable in some cases.