A Complete Classification on the Center-Focus Problem of a Generalized Cubic Kukles System with a Nilpotent Singular Point

• Feng Li ^[1] ; Ting Chen ^[3] ; Yuanyuan Liu ^[1] ; Pei Yu ^[2]
  1. [1] Linyi University

     Linyi University

     China

     
  2. [2] Western University

     Western University

     Azerbaiyán

     
  3. [3] Guangdong University of Finance and Economics

  Mostrar afiliaciones +
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 1, 2024
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • In this paper, we study the center-focus problem for a generalized cubic Kukles system with a nilpotent singular point, which consists of a cubic system with an extra 4th-order term. A complete classification is given on the center conditions which are explicitly expressed in term of the system parameters. A total of 15 cases are obtained, among them 4 for the generalized cubic Kukles system and 12 for the cubic Kukles system, with one common for both. One of the center conditions is analytic. Moreover, it is shown that 8 small-amplitude limit cycles can bifurcate in the neighborhood of the singular point for the generalized cubic Kukles system, while only 7 small-amplitude limit cycles can exist around the singular point for the cubic Kukles system. The center-focus problem for the generalized cubic Kukles system with a nilpotent origin is thoroughly solved