The period function of classical Lienard equations

In this paper we study the number of critical points that the period function of a center of a classical Lienard equation can have. Centers of classical Lienard equations are related to scalar differential equations x + x + f(x)x = 0, 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 e, can be reduced to the study of slow-fast Lienard 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 Lienard 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. (c) 2006 Elsevier Inc. All rights reserved.