The number of limit cycles due to polynomial perturbations of the harmonic oscillator

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.