数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (3): 1169-1187.doi: 10.1007/s10473-025-0322-4

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HOPF BIFURCATION PROBLEM BY PERTURBING A CLASS OF QUARTIC LINEAR-LIKE HAMILTONIAN SYSTEMS

Yanqin XIONG, Guangping HU   

  1. School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
  • 收稿日期:2023-02-23 修回日期:2024-01-01 出版日期:2025-05-25 发布日期:2025-09-30

HOPF BIFURCATION PROBLEM BY PERTURBING A CLASS OF QUARTIC LINEAR-LIKE HAMILTONIAN SYSTEMS

Yanqin XIONG, Guangping HU   

  1. School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
  • Received:2023-02-23 Revised:2024-01-01 Online:2025-05-25 Published:2025-09-30
  • Contact: Yanqin XIONG, E-mail: yqxiong@nuist.edu.cn
  • About author:Guangping HU, E-mail: hugp@nuist.edu.cn
  • Supported by:
    National Natural Science Foundations of China (12371171) and the Natural Science Foundation of Jiangsu Province (BK20221339).

摘要: We study the limit cycle bifurcations perturbing a class of quartic linear-like Hamiltonian systems having an elementary center at the origin. First, using methods of the qualitative theory, all possible phase portraits of the unperturbed system are found. Then, using the first order Melnikov function, Hopf bifurcation problem of the perturbed system is investigated, and an upper bound for the function is obtained near the origin.

关键词: quartic near-Hamiltonian system, phase portrait, Hopf bifurcation, Hopf cyclicity

Abstract: We study the limit cycle bifurcations perturbing a class of quartic linear-like Hamiltonian systems having an elementary center at the origin. First, using methods of the qualitative theory, all possible phase portraits of the unperturbed system are found. Then, using the first order Melnikov function, Hopf bifurcation problem of the perturbed system is investigated, and an upper bound for the function is obtained near the origin.

Key words: quartic near-Hamiltonian system, phase portrait, Hopf bifurcation, Hopf cyclicity

中图分类号: 

  • 34C05