Polynomial slow-fast systems on the Poincaré–Lyapunov sphere

Abstract

The main goal of this paper is to study compactifications of polynomial slow-fast systems. More precisely, the aim is to give conditions in order to guarantee normal hyperbolicity at infinity of the Poincaré–Lyapunov sphere for slow-fast systems defined in Rn. For the planar case, we prove a global version of the Fenichel Theorem, which assures the persistence of invariant manifolds in the whole Poincaré–Lyapunov disk. We also discuss the occurrence of non normally hyperbolic points at infinity, namely: fold, transcritical and pitchfork singularities.