On the analytic integrability of the Liénard analytic differential systems

Jaume  Llibre; Claudia  Valls

doi:10.3934/dcdsb.2016.21.557

Article Contents

2016, Volume 21, Issue 2: 557-573. Doi: 10.3934/dcdsb.2016.21.557

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On the analytic integrability of the Liénard analytic differential systems

Jaume Llibre^1, and
Claudia Valls^2,

1.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia
2.
Departamento de Matemática, Instituto Superior Técnico , Universidade Técnica de Lisboa, Av. Rovisco Pais 1049-001, Lisboa

Received: June 30, 2014

Revised: August 31, 2015

Published: February 2016

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Abstract

We consider the Liénard analytic differential systems $\dot x = y$, $\dot y =-g(x) -f(x)y$, where $f,g: \mathbb{R}\to \mathbb{R}$ are analytic functions and the origin is an isolated singular point. Then for such systems we characterize the existence of local analytic first integrals in a neighborhood of the origin and the existence of global analytic first integrals.

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Mathematics Subject Classification: Primary: 34C05, 34A34, 34C14.

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