Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the local analytic integrability at the singular point of a class of Liénard analytic differential systems

Authors: Jaume Llibre and Clàudia Valls
Journal: Proc. Amer. Math. Soc. 138 (2010), 253-261
MSC (2000): Primary 34C05, 34A34, 34C14
Published electronically: August 19, 2009
MathSciNet review: 2550190
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the Liénard analytic differential systems $ \dot x = y$, $ \dot y= -cx -f(x)y$, with $ c \in \mathbb{R}$ and $ f: \mathbb{R}\to \mathbb{R}$ an analytic function. Then for such systems we characterize the existence of local analytic first integrals in a neighborhood of the singular point located at the origin.

References [Enhancements On Off] (What's this?)

  • 1. C. CHRISTOPHER, An algebraic approach to the classification of centers in polynomial Liénard systems, J. Math. Anal. Appl. 229 (1999), 319-329. MR 1664344 (99k:34056)
  • 2. A. CIMA, A. GASULL AND F. MA˜NOSAS, Cyclicity of a family of vector fields, J. Math. Anal. Appl. 196 (1995), 921-937. MR 1365231 (96j:58143)
  • 3. S.D. FURTA, On non-integrability of general systems of differential equations, Z. Angew Math. Phys. 47 (1996), 112-131. MR 1408674 (97f:34003)
  • 4. P. JOYAL AND C. ROUSSEAU, Saddle quantities and applications, J. Diff. Equations 78 (1989), 374-399. MR 992152 (90b:58225)
  • 5. M.A. LIAPUNOV, Problème général de la stabilité du mouvement, Ann. of Math. Stud., 17, Princeton University Press; Oxford University Press, London, 1947. MR 0021186 (9:34j)
  • 6. R. MOUSSU, Une démonstration géométrique d'un théorème de Lyapunov-Poincaré, Astérisque 98-99 (1982), 216-223. MR 724449 (85g:58012)
  • 7. H. POINCARÉ, Mémoire sur les courbes définies par les équations différentielles, Journal de Mathématiques 37 (1881), 375-422; Oeuvres de Henri Poincaré, vol. I, Gauthier-Villars, Paris, 1951, pp. 3-84.
  • 8. H. POINCAR´E, Sur l'intégration des équations différentielles du premier order et du premier degré I and II, Rendiconti del circolo matematico di Palermo 5 (1891), 161-191; 11 (1897), 193-239.
  • 9. C. ZUPPA, Order of cyclicity of the singular point of Liénard's polynomial vector fields, Bol. Soc. Brasil. Mat. 12 (1981), 105-111. MR 688192 (84h:58126)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 34C05, 34A34, 34C14

Retrieve articles in all journals with MSC (2000): 34C05, 34A34, 34C14

Additional Information

Jaume Llibre
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bella- terra, Barcelona, Catalonia, Spain

Clàudia Valls
Affiliation: Departamento de Matemática, Instituto Superior Técnico, 1049–001 Lisboa, Portugal

Keywords: Analytic integrability, local analytic integrability, Li\'{e}nard differential system
Received by editor(s): February 21, 2009
Received by editor(s) in revised form: April 30, 2009
Published electronically: August 19, 2009
Communicated by: Yingfei Yi
Article copyright: © Copyright 2009 American Mathematical Society

American Mathematical Society