An Improved Lower Bound on the Number of Limit Cycles Bifurcating from a Hamiltonian Planar Vector Field of Degree 7

The limit cycle bifurcations of a Z2 equivariant planar Hamiltonian vector field of degree 7 under Z2 equivariant degree 7 perturbation is studied. We prove that the given system can have at least 53 limit cycles. This is an improved lower bound for the weak formulation of Hilbert’s 16th problem for degree 7, i.e., on the possible number of limit cycles that can bifurcate from a degree 7 planar Hamiltonian system under degree 7 perturbation,