A Note on a Result of G. S. Petrov About the Weakened 16th Hilbert Problem

G. S. Petrov [Functional Anal. Appl. 22 (1988), 72-73] and P. Mardesic [Ergodic Theory Dynamical Systems 10 (1990), 523-529] have proved that for system ẋ = y + ϵP(x, y), ẏ = 1 − 3x2 + ϵQ(x, y), where P(x, y) and Q(x, y) are polynomials of x, y with degree ≤ N, if the first order Melnikov function M1(h) ≢ 0, then the lowest upper bound B(N) of the number of limit cycles of the above system is N − 1. We prove that if M1(h) ≡ 0 and the second order Melnikov function M2(h) ≢ 0, then [formula]