Quadratic slow-fast systems on the plane

Abstract

In this paper singularly perturbed quadratic polynomial differential systems εẋ=Pε(x,y)=P(x,y,ε),ẏ=Qε(x,y)=Q(x,y,ε)with x,y∈R,ε⩾0 and (Pε,Qε)=1 for ε>0, are considered. We prove that there are 10 classes of equivalence for these systems. We describe the dynamics of these 10 classes on the Poincaré disc when ε=0. For ε>0, we present the possible local behavior of the solutions near of a finite and infinite equilibrium point under suitable conditions. More specifically, if p0 is a finite equilibrium point then we obtain the local behavior for ε>0 using Fenichel theory. Assuming that p0 is an infinite equilibrium point, there exists K⊂M0 normally hyperbolic and p0∈M0′∩K using the Poincaré compactification and algebraic invariant we describe globally the dynamics for ε>0 small of some classes of equivalence.