Equivalence of two Melnikov functions (Q1919024)

The author studies the system \(x'= F(x)+ \alpha G(x,\delta)\), where \(x\in \mathbb{R}^3\), \(\delta\in \mathbb{R}^k\), and \(\alpha\in \mathbb{R}\) is a small parameter. It is assumed that for \(\alpha =0\) the system has a homoclinic trajectory \(L\) of a hyperbolic rest point. A new form of the Melnikov functions for \(L\) is obtained. It is shown that this new Melnikov function is equivalent to the standard one.