Limit cycles of perturbations of quadratic Hamiltonian vector fields (Q1900210)

We find an upper bound (three) for the number of limit cycles of the perturbed plane Hamiltonian system \(dH+ \varepsilon (P (x,y) dx+ Q(x, y) dy) =0\) for arbitrary quadratic polynomials \(P\), \(Q\) and cubic Hamiltonians \(H(x, y)\) having the following properties --- the system \(dH=0\) has three saddle points and one center; --- the corresponding critical values of \(H\) are all different.