New configurations of 24 limit cycles in a quintic system

This paper concerns with the number and distributions of limit cycles in a Z3Z3-equivariant quintic planar polynomial system. 24 limit cycles are found in this system and two different configurations of them are shown by combining the methods of double homoclinic loops bifurcation, Poincaré bifurcation and qualitative analysis. The two configurations of 24 limit cycles obtained in this paper are new. The results obtained are useful to the study of weakened 16th Hilbert Problem.