Bifurcations of the limit cycles in a z3-equivariant quartic planar vector field

In this paper, a Z3-equivariant quartic planar vector field is studied. The Hopf bifurcation method and polycycle bifurcation method are combined to study the limit cycles bifurcated from the polycycle with three hyperbolic saddle points. It is found that this special quartic planar polynomial system has at least three large limit cycles which surround 13 singular points. By applying the double homoclinic loops bifurcation method and Poincaré–Bendixson theorem, we conclude that 16 limit cycles with two different configurations exist in this special planar polynomial system. The results acquired in this paper are useful to the study of the second part of 16th Hilbert Problem.