Transversal heteroclinic orbits in general degenerate cases (Q1919040)

The author studies the system \(x'= f(x)+ \varepsilon g(t,x, \varepsilon, \mu)\), where \(x\in \mathbb{R}^n\), \(\mu\in \mathbb{R}^m\), and \(\varepsilon \in\mathbb{R}\) is a small parameter. It is assumed that for \(\varepsilon =0\) the system has two hyperbolic rest points \(p,q\), and a \(d\)-dimensional manifold of trajectories going from \(p\) to \(q\). A special Melnikov vector is constructed. In terms of this vector, the author describes the local structure of the set \(\{\mu\}\) which corresponds to the existence of transversal heteroclinic trajectories or of tangential heteroclinic trajectories for small \(\varepsilon\). These results generalize well-known statements by Hale, Moser, Palmer, and other authors.