The period function of classical Liénard equations |
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Authors: | P. De Maesschalck |
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Affiliation: | Universiteit Hasselt, Campus Diepenbeek, Agoralaan-Gebouw D, B-3590 Diepenbeek, Belgium |
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Abstract: | 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 2?−1. We show that the existence of a finite upperbound on the number of critical periods, only depending on the value of ?, 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 2?−1 the number of critical periods is at most 2?−2. We show the occurrence of slow-fast Liénard systems exhibiting 2?−2 critical periods, elucidating a qualitative process behind the occurrence of critical periods. It all provides evidence for conjecturing that 2?−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. |
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Keywords: | 34C25 34C26 34E15 34E10 |
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