Resumen de Perturbation of a Period Annulus Bounded by a Heteroclinic Loop Connecting Two Hyperbolic Saddles

Xianbo Sun

• In this paper, we study the bound of the number of isolated zeros of the Abelian integral I(h, δ) associated to system x˙ = y, y˙ = −x(x2 − 1)(x2 + 2)2 under the perturbations (α0 + α1x2 + α2x4 + α3x6)y ∂ ∂y , where 0 < || 1 and αi ∈ R, i = 0, 1, 2, 3. The period annulus of the unperturbed system is bounded by a two-saddle loop surrounding an elementary center. We divide the parameter space {α0, α1, α2, α3} into four parts, for each case the least upper bound of number of zeros of I(h, δ) will be investigated. We find that a smaller upper bound can be obtained when four generating elements of I(h, δ) are reduced to three special ones. Moreover, four zeros of I(h, δ) can be reached under case 4 by the asymptotic expansions of I(h, δ) and zero bifurcation theory.