Bifurcations of limit cycles in a \(Z_8\)-equivariant planar vector field of degree 7 (Q1763031)

The authors consider the weakened Hilbert's 16th problem for symmetric planar perturbed polynomial Hamiltonian systems. They construct an example of such system which is \(Z_8\)-equivariant. By using bifurcation theory of planar polynomial dynamical systems and the method of detection functions, with the help of numerical analysis, the authors show that there exist parameter groups such that the perturbed polynomial Hamiltonian vector field of degree~\(7\) has at least \(7^2=49\) limit cycles with \(Z_8\)~symmetry.