Abstract
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 quasi-homogeneous 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.
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Zhuang, Z., Liu, C. Critical Periods of the Sum of Two Quasi-Homogeneous Hamiltonian Vector Fields. Qual. Theory Dyn. Syst. 22, 90 (2023). https://doi.org/10.1007/s12346-023-00786-z
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DOI: https://doi.org/10.1007/s12346-023-00786-z