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On Massera’s theorem concerning the uniqueness of a periodic solution for the Liénard equation. When does such a periodic solution actually exist?

Lilia Rosati and Gabriele Villari*

Author Affiliations

Dipartimento di Matematica ‘U. Dini’, Università degli Studi di Firenze, viale Morgagni 67/A, Firenze, 50134, Italy

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Boundary Value Problems 2013, 2013:144  doi:10.1186/1687-2770-2013-144

Published: 11 June 2013

Abstract

In this note we consider the classical Massera theorem, which proves the uniqueness of a periodic solution for the Liénard equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M1">View MathML</a>

and investigate the problem of the existence of such a periodic solution when f is monotone increasing for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M2">View MathML</a> and monotone decreasing for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/144/mathml/M3">View MathML</a> but with a single zero, because in this case the existence is not granted. Sufficient conditions for the existence of a periodic solution and also a necessary condition, which proves that with this assumptions actually it is possible to have no periodic solutions, are presented.

MSC: 34C05, 34C25.

Keywords:
Liénard equation; limit cycles; Massera theorem