Journal of Guangxi Normal University(Natural Science Edition) ›› 2022, Vol. 40 ›› Issue (6): 109-115.doi: 10.16088/j.issn.1001-6600.2021102002

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New Lower Bounds of Limit Cycles for a Class of Three-dimensional Cubic Systems

LIU Jukun1, HUANG Wentao2*, LIU Hongpu1   

  1. 1. School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin Guangxi 541004, China;
    2. School of Mathematics and Statistics, Guangxi Normal University, Guilin Guangxi 541006, China
  • Received:2021-10-20 Revised:2021-12-07 Online:2022-11-25 Published:2023-01-17

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Abstract: The center and limit cycle bifurcation of a class of three-dimensional cubic systems with two symmetric singularities are studied. Firstly, the first eight singular points of the adjoint complex system are calculated with the help of computer algebra software, a set of necessary conditions for the two singular points to become the center are obtained, and its sufficiency is further proved. Then the condition that the two singularities become the 8th order fine focus at the same time is derived. Finally, by using the Jacobian determinant method, it is proved that there are at least 16 small amplitude limit cycles in the system, and a new lower bound for the number of limit cycles of three-dimensional cubic system is given.

Key words: three-dimensional cubic system, singularity quantity, central manifold, center, limit cycle

CLC Number: 

  • O175.12
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