A note on a result of G. S. Petrov about the weakened 16th Hilbert problem (Q1892545)

\textit{G. S. Petrov} [Funct. Anal. Appl. 22, 72-73 (1988); translation from Funkts. Anal. Prilozh. 22, 83-84 (1988; Zbl 0645.33003)] and \textit{P. Mardesic} [Ergodic Theory Dyn. Syst. 10, 523-529 (1990; Zbl 0706.58054)] have proved that for system \(\dot x = y + \varepsilon P(x,y)\), \(\dot y = 1 - 3x^2 + \varepsilon Q(x,y)\), where \(P(x,y)\) and \(Q(x,y)\) are polynomials of \(x,y\) with degree \(\leq N\), if the first order Melnikov function \(M_1 (h) \not \equiv 0\), then the lowest upper bound \(B(N)\) of the number of limit cycles of the above system is \(N - 1\). We prove that if \(M_1(h) \equiv 0\) and the second order Melnikov function \(M_2(h) \not \equiv 0\), then \[ B(N) = \begin{cases} 2N - 2, \quad & N \text{- even},\\ 2N - 3, \quad & N \text{-odd}. \end{cases} . \]