Quartic Liénard Equations with Linear Damping

Abstract

In this paper we prove that the quartic Liénard equation with linear damping {x˙=y,y˙=−(a0+x)y−(b0+b1x+b2x2+b3x3+x4)} can have at most two limit cycles, for the parameters kept in a small neighborhood of the origin (a0,b0,b1,b2,b3)=(0,0,0,0,0) . Near the origin in the parameter space, the Liénard equation is of singular type and we use singular perturbation theory and the family blow up. To study the limit cycles globally in the phase space we need a suitable Poincaré–Lyapunov compactification.