The period function of classical Liénard equations

In this paper we study the number of critical points that the period function of a center of a classical Liénard equation can have. Centers of classical Liénard equations are related to scalar differential equations , with f an odd polynomial, let us say of degree 2ℓ−1. We show that the existence of a finite upperbound on the number of critical periods, only depending on the value of ℓ, can be reduced to the study of slow–fast Liénard equations close to their limiting layer equations. We show that near the central system of degree 2ℓ−1 the number of critical periods is at most 2ℓ−2. We show the occurrence of slow–fast Liénard systems exhibiting 2ℓ−2 critical periods, elucidating a qualitative process behind the occurrence of critical periods. It all provides evidence for conjecturing that 2ℓ−2 is a sharp upperbound on the number of critical periods. We also show that the number of critical periods, multiplicity taken into account, is always even.