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March  2016, 21(2): 557-573. doi: 10.3934/dcdsb.2016.21.557

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  July 2014 Revised  September 2015 Published  November 2015

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.
Keywords: Formal integrability, analytic integrability, Liénard system..
Mathematics Subject Classification: Primary: 34C05, 34A34, 34C1.
Citation: Jaume Llibre, Claudia Valls. On the analytic integrability of the Liénard analytic differential systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 557-573. doi: 10.3934/dcdsb.2016.21.557
References:
[1]

C. Christopher, An algebraic approach to the classification of centers in polynomial Liénard systems, J. Math. Anal. Appl., 229 (1999), 319-329. doi: 10.1006/jmaa.1998.6175.  Google Scholar

[2]

C. Christopher and N. G. Lloyd, Small-amplitude limit cycles in polynomial Liénard systems, NoDEA, 3 (1996), 183-190. doi: 10.1007/BF01195913.  Google Scholar

[3]

H. Dulac, Détermination et intégration d'une certaine classe d'équation différentielles ayant pour point singulier un centre, Bull. Sciences Math. Sér. 2, 32 (1908), 230-252. Google Scholar

[4]

S. D. Furta, On non-integrability of general systems of differential equations, Z. angew Math. Phys., 47 (1996), 112-131. doi: 10.1007/BF00917577.  Google Scholar

[5]

W. Li, J. Llibre and X. Zhang, Local first integrals of differential systems and diffeomorphisms, Z. angew. Math. Phys., 54 (2003), 235-255. doi: 10.1007/s000330300003.  Google Scholar

[6]

A. M. Liapunov, Stability of Motion, Mathematics in Science and Engineering, Vol. 30 Academic Press, New York-London, 1966. doi: 10.1002/zamm.19680480223.  Google Scholar

[7]

J. Llibre and D. Schlomiuk, The geometry of quadratic differential systems with a weak focus} {of third order, Canadian J. Math., 56 (2004), 310-343. doi: 10.4153/CJM-2004-015-2.  Google Scholar

[8]

R. Moussu, Une démonstration d'un théorème de Lyapunov-Poincaré, Astérisque, 98/99 (1982), 216-223.  Google Scholar

[9]

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. Google Scholar

[10]

H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, J. Math. Pure Appl., (4) 1 (1885), 167-244. Google Scholar

[11]

H. Poincaré, 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. Google Scholar

[12]

D. Schlomiuk, Algebraic and geometric aspects of the theory of polynomial vector fields, Bifurcation and Periodic Orbits of Vector Fields, (ed. D. Schlomiuk) 408 (1993), 429-467. doi: 10.1007/978-94-015-8238-4_10.  Google Scholar

[13]

S. Songling, A method of constructing cycles without contact around a weak focus, J. Differential Equations, 41 (1981), 301-312. doi: 10.1016/0022-0396(81)90039-5.  Google Scholar

[14]

S. Songling, On the structure of Poincaré-Lyapunov constants for the weak focus of polynomial vector fields, J. Differential Equations, 52 (1984), 52-57. doi: 10.1016/0022-0396(84)90133-5.  Google Scholar

[15]

C. Zuppa, Order of cyclicity of the singular point of Liénard's polynomial vector fields, Bol. Soc. Brasil Mat., 12 (1982), 105-111. doi: 10.1007/BF02584662.  Google Scholar

show all references

References:
[1]

C. Christopher, An algebraic approach to the classification of centers in polynomial Liénard systems, J. Math. Anal. Appl., 229 (1999), 319-329. doi: 10.1006/jmaa.1998.6175.  Google Scholar

[2]

C. Christopher and N. G. Lloyd, Small-amplitude limit cycles in polynomial Liénard systems, NoDEA, 3 (1996), 183-190. doi: 10.1007/BF01195913.  Google Scholar

[3]

H. Dulac, Détermination et intégration d'une certaine classe d'équation différentielles ayant pour point singulier un centre, Bull. Sciences Math. Sér. 2, 32 (1908), 230-252. Google Scholar

[4]

S. D. Furta, On non-integrability of general systems of differential equations, Z. angew Math. Phys., 47 (1996), 112-131. doi: 10.1007/BF00917577.  Google Scholar

[5]

W. Li, J. Llibre and X. Zhang, Local first integrals of differential systems and diffeomorphisms, Z. angew. Math. Phys., 54 (2003), 235-255. doi: 10.1007/s000330300003.  Google Scholar

[6]

A. M. Liapunov, Stability of Motion, Mathematics in Science and Engineering, Vol. 30 Academic Press, New York-London, 1966. doi: 10.1002/zamm.19680480223.  Google Scholar

[7]

J. Llibre and D. Schlomiuk, The geometry of quadratic differential systems with a weak focus} {of third order, Canadian J. Math., 56 (2004), 310-343. doi: 10.4153/CJM-2004-015-2.  Google Scholar

[8]

R. Moussu, Une démonstration d'un théorème de Lyapunov-Poincaré, Astérisque, 98/99 (1982), 216-223.  Google Scholar

[9]

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. Google Scholar

[10]

H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, J. Math. Pure Appl., (4) 1 (1885), 167-244. Google Scholar

[11]

H. Poincaré, 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. Google Scholar

[12]

D. Schlomiuk, Algebraic and geometric aspects of the theory of polynomial vector fields, Bifurcation and Periodic Orbits of Vector Fields, (ed. D. Schlomiuk) 408 (1993), 429-467. doi: 10.1007/978-94-015-8238-4_10.  Google Scholar

[13]

S. Songling, A method of constructing cycles without contact around a weak focus, J. Differential Equations, 41 (1981), 301-312. doi: 10.1016/0022-0396(81)90039-5.  Google Scholar

[14]

S. Songling, On the structure of Poincaré-Lyapunov constants for the weak focus of polynomial vector fields, J. Differential Equations, 52 (1984), 52-57. doi: 10.1016/0022-0396(84)90133-5.  Google Scholar

[15]

C. Zuppa, Order of cyclicity of the singular point of Liénard's polynomial vector fields, Bol. Soc. Brasil Mat., 12 (1982), 105-111. doi: 10.1007/BF02584662.  Google Scholar

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