We prove that in quadratic perturbations of generic quadratic Hamiltonian vector fields with three saddle points and one centre there can appear at most two limit cycles. This bound is exact.

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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 fieldṡ z = iz − i(z +z) 3 andż = iz +z 2 , 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.

An explicit upper bound Z(3, n) 5n+15 is derived for the number of the zeros of the integral h → I (h) = H =h g(x, y) dx −f (x, y) dy of degree n polynomials f, g, on the open interval for which the cubic curve {H = h} contains an oval. The proof exploits the properties of the Picard-Fuchs system satisfied by the four basic integrals H

We consider arbitrary polynomial perturbationsformula hereof the harmonic oscillator. In (1), f and g are polynomials of x, y with coefficients depending analytically on the small parameter ε. Let us denote n = max (deg f, deg g), H = ½(x2 + y2). Using the energy level H = h as a parameter, we can express the first return mapping of (1) in terms of h and ε. For the corresponding displacement function d(h, ε) = [Pscr ](h, ε)−h we obtain the following representation as a power series in ε:formula herewhich is convergent for small ε. The Melnikov functions Mk(h) are defined for h[ges ]0. Each isolated zero h0∈ (0, ∞) of the first non-vanishing coefficient in (2) corresponds to a limit cycle of (1) emerging from the circle x2 + y2 = 2h0 when ε increases from zero. Our main result in this paper is the following.

The paper derives a formula for the second variation of the displacement function for polynomial perturbations of Hamiltonian systems with elliptic or hyperelliptic Hamiltonians H(x, y) l " # y#kU(x) in terms of the coefficients of the perturbation. As an application, the conjecture stated by C. Chicone that a specific cubic system appearing in a deformation of singularity with two zero eigenvalues has at most two limit cycles is proved.

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with analytic * j (=)=O(=), have at most two limit cycles that bifurcate for small ={0 from any period annulus of the unperturbed system. This fact agrees with previous results of Petrov, Dangelmayr and Guckenheimer, and Chicone and Iliev, but shows that the result of three limit cycles for the asymmetrically perturbed, exterior Duffing oscillator, recently obtained by Jebrane and Z 4 o*a dek, is incorrect. The proofs follow by deriving an explicit formula for the kth-order Melnikov function, M k (h), and using a Picard Fuchs analysis to show that, in each case, M k (h) has at most two zeros. Moreover, the method developed in this paper for determining the higher-order Melnikov functions also applies to more general perturbations of these systems. Academic Press