Abstract

In this paper, we consider a small polynomial perturbation of the Hamiltonian vector field with the Hamiltonian F(x, y) = x [y2 - (x - 3)2] having a center bounded by a triangle. The main result of this work is that the principal Poincaré - Pontryagin function associated with such a perturbation and with the family of ovals surrounding the center belongs to the ℂ[t, 1/t] module generated by Abelian integrals I0(t) and I2(t) and by I*(t), where I*(t) is not an Abelian integral. We show that, in general, the principal Poincaré - Pontryagin function of order two of a polynomial perturbation of the degree at least five is not an Abelian integral. © 2006 Springer Science+Business Media, Inc.