Generalization of Mel'nikov's theorem to separatrix contours in the autonomous case (Q1357927)

Consider the system \[ \dot x=P(x,y, \mu), \quad \dot y=Q(x,y, \mu), \tag{1} \] in the plane, where \(\mu\in \mathbb{R}^n\) is a parameter, and \(P,Q\in C^{(r)}\) \((r>1)\). Suppose that for \(\mu=0\) the system has two hyperbolic saddles \(z_1\) and \(z_2\), and a separatrix \(L_{12}\) joining \(z_1\) and \(z_2\). The authors obtain Mel'nikov-type conditions for the existence in \(\mathbb{R}^n\) of a local bifurcation manifold which corresponds to the system's having separatrices \(L_{12} (\mu)\) joining saddles \(z_1(\mu)\) and \(z_2(\mu)\) (with \(z_1(0) =z_1\), \(z_2(0) =z_2\) and \(L_{12} (0)= L_{12})\). They also consider the case for \(k\geq 1\) separatrices joining \(q\geq 1\) saddles.