Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems

We study degree n polynomial perturbations of quadratic reversible Hamiltonian vector fields with one center and one saddle point. It was recently proved that if the first Poincaré–Pontryagin integral is not identically zero, then the exact upper bound for the number of limit cycles on the finite plane is n− 1. In the present paper we prove that if the first Poincaré–Pontryagin function is identically zero, but the second is not, then the exact upper bound for the number of limit cycles on the finite plane is 2(n−1). In the case when the perturbation is quadratic (n = 2) we obtain a complete result—there is a neighborhood of the initial Hamiltonian vector field in the space of all quadratic vector fields, in which any vector field has at most two limit cycles.