The displacement map associated to polynomial unfoldings of planar Hamiltonian vector fields

We study the displacement map associated to small one-parameter polynomial unfoldings of polynomial Hamiltonian vector fields on the plane. Its leading term, the generating function M(t), has an analytic continuation in the complex plane and the real zeroes of M(t) correspond to the limit cycles bifurcating from the periodic orbits of the Hamiltonian flow. We give a geometric description of the monodromy group of M(t) and use it to formulate sufficient conditions for M(t) to satisfy a differential equation of Fuchs or Picard-Fuchs type. As examples, we consider in more detail the Hamiltonian vector fields ż = iz̄ − i(z + z̄)3 and ż = iz + z̄2, possessing a rotational symmetry of order two and three, respectively. In both cases M(t) satisfies a Fuchs-type equation but in the first example M(t) is always an Abelian integral (that is to say, the corresponding equation is of Picard-Fuchs type) while in the second one this is not necessarily true. We derive an explicit formula of M(t) and estimate the number of its real zeroes.