Second-order analysis in polynomially perturbed reversible quadratic Hamiltonian systems

We study degree n polynomial perturbations of quadratic reversible Hamilton ian vector fields with one center and one saddle point. It was recently pro ved that if the first Poincare-Pontryagin integral is not identically zero, then the exact upper bound for the number of Limit cycles on the finite pl ane is n - 1. In the present paper we prove that if the first Poincare-Pont ryagin 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 complet e 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.