Bifurcation to homoclinic connections of the focus-saddle type (Q1076890)

The essential hypothesis of Sil'nikov's theorem is the existence of a homoclinic connection of the focus-saddle type, which implies a complicated configuration of the invariant manifold of the hyperbolic fixed point \(\theta\). Such a hypothesis is hard to verify. In this paper we develop some techniques to construct parametric families of regular vector fields which will have the above connection for parameter values belonging to a manifold of codimension one. The equation of that manifold will be defined later.