Bifurcations of limit cycles in a Z4-equivariant planar polynomial vector field of degree 7

The present paper intends to explore the number and distribution of limit cycles for a class of perturbed Z4-equivariant planar Hamiltonian system of degree 7. The bifurcation theory of planar dynamical systems and the method of detection functions are applied to study the given perturbed system. By controlling the perturbation parameters, an example of a special perturbed Z4-equivariant vector field having 41 limit cycles is found. The distribution of the above 41 limit cycles is also given. The results are significant to the study of weaken Hilbert's 16th problem and to the acquaintance of the characteristics of higher-order equivariant vector fields.