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

• Ziwei Zhuang ^[1] ; Changjian Liu ^[1]
  1. [1] Sun Yat-sen University

     Sun Yat-sen University

     China

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 22, Nº 3, 2023
• Idioma: inglés
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• Resumen

  • 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.