Semi-local normal forms of saddle connections and a study of non-elementary singularities

Abstract

We develop tools to study the transition in planar vector fields near non-elementary singularities. In particular we focus on symmetric connections which appear naturally in blow-up phase portraits and the circle at infinity. First we develop a semi-local normal form which is applicable near the connection. Then we add some non-smooth variables such that we can normally linearize the system. This allows us to express the transition map near the connection in terms of a finite amount of variables and we provide an asymptotic expansion of the transition map. Finally, we apply the results to homoclinic connections containing a non-elementary singularity, in this case a cusp or a fake saddle.