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

On the analytic integrability of the Liénard analytic differential systems

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.
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.  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.  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.   Google Scholar

[4]

S. D. Furta, On non-integrability of general systems of differential equations,, Z. angew Math. Phys., 47 (1996), 112.  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.  doi: 10.1007/s000330300003.  Google Scholar

[6]

A. M. Liapunov, Stability of Motion,, Mathematics in Science and Engineering, 30 (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.  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.   Google Scholar

[9]

H. Poincaré, Mémoire sur les courbes définies par les équations différentielles,, Journal de Mathématiques, 37 (1881), 375.   Google Scholar

[10]

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

[12]

D. Schlomiuk, Algebraic and geometric aspects of the theory of polynomial vector fields,, Bifurcation and Periodic Orbits of Vector Fields, 408 (1993), 429.  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.  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.  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.  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.  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.  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.   Google Scholar

[4]

S. D. Furta, On non-integrability of general systems of differential equations,, Z. angew Math. Phys., 47 (1996), 112.  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.  doi: 10.1007/s000330300003.  Google Scholar

[6]

A. M. Liapunov, Stability of Motion,, Mathematics in Science and Engineering, 30 (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.  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.   Google Scholar

[9]

H. Poincaré, Mémoire sur les courbes définies par les équations différentielles,, Journal de Mathématiques, 37 (1881), 375.   Google Scholar

[10]

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

[12]

D. Schlomiuk, Algebraic and geometric aspects of the theory of polynomial vector fields,, Bifurcation and Periodic Orbits of Vector Fields, 408 (1993), 429.  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.  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.  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.  doi: 10.1007/BF02584662.  Google Scholar

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