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

Abstract

This paper focuses on investigating the bifurcation of limit cycles and centers within a specific class of three-dimensional cubic systems possessing Z3-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 Z3-equivariant symmetry.

Keywords

• three-dimensional cubic systems
• Z3-equivariant symmetry
• limit cycle
• center
• Darboux integral method