ABELIAN INTEGRALS FOR THE ONE-PARAMETER BOGDANOV–TAKENS SYSTEM

Abstract

An explicit upper bound Z(2, n) ≤ n + m - 1 is derived for the number of zeros of Abelian integrals M1(h) = ∮γ(h) P(x, y) dy - Q(x, y) dx on the open interval (0, 1/6), where γ(h) is an oval lying on the algebraic curve Hλ = (1/2)x2 + (1/2)y2 - (1/3)x3 - λy3 = h, P(x, y), Q(x, y) are polynomials of x and y, and max { deg P(x, y), deg Q(x, y)} = n. The proof exploits the expansion of the first order Melnikov function M1(h, λ) near λ = 0 and assume (∂m/∂λm)M1(h, λ)|λ = 0 not vanish identically.