Bifurcation of small limit cycles in Z5-equivariant planar vector fields of order 5

In this paper, we consider bifurcation of small limit cycles from Hopf-type singular points in Z5-equivariant planar vector fields of order 5. We apply normal form theory and the technique of solving coupled multivariate polynomial equations to prove that the maximal number of small limit cycles that such vector fields can have is 25. In addition, we show that no large limit cycles exist. Thus, H(5)⩾25, where H(n) denotes the Hilbert number of the nth-degree polynomial vector fields. This improves the best result of H(5)⩾24 existing in the current literature.