Bifurcation of Limit Cycles in Z10-Equivariant Vector Fields of Degree 9

In this paper, we consider the weakened Hilbert’s 16th problem for symmetric planar perturbed polynomial Hamiltonian systems. In particular, a perturbed Hamiltonian polynomial vector field of degree 9 is studied, and an example of Z10-equivariant planar perturbed Hamiltonian systems is constructed. With maximal number of closed orbits, it gives rise to different configurations of limit cycles. By applying the bifurcation theory of planar dynamic systems and the method of detection functions, with the aid of numerical simulations, we show that a polynomial vector field of degree 9 with Z10 symmetry can have at least 80 limit cycles, i.e. H(9) ≥ 9 − 1.