Bifurcation of Limit Cycles and Center in 3D Cubic Systems with Z 3 -Equivariant Symmetry

Ting Huang (), Jieping Gu, Yuting Ouyang and Wentao Huang
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Ting Huang: School of Computer Engineering, Guangzhou City University of Technology, Guangzhou 510800, China
Jieping Gu: School of Education, Guangxi Vocational Normal University, Nanning 530007, China
Yuting Ouyang: College of Mathematics and Statistics, Guangxi Normal University, Guilin 541006, China
Wentao Huang: College of Mathematics and Statistics, Guangxi Normal University, Guilin 541006, China

Mathematics, 2023, vol. 11, issue 11, 1-22

Abstract: This paper focuses on investigating the bifurcation of limit cycles and centers within a specific class of three-dimensional cubic systems possessing Z 3 -equivariant symmetry. By calculating the singular point values of the systems, we obtain a necessary condition for a singular point to be a center. Subsequently, the Darboux integral method is employed to demonstrate that this condition is also sufficient. Additionally, we demonstrate that the system can bifurcate 15 small amplitude limit cycles with a distribution pattern of 5 − 5 − 5 originating from the singular points after proper perturbation. This finding represents a novel contribution to the understanding of the number of limit cycles present in three-dimensional cubic systems with Z 3 -equivariant symmetry.

Keywords: three-dimensional cubic systems; Z3-equivariant symmetry; limit cycle; center; Darboux integral method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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