American Institute of Mathematical Sciences

Advanced
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 and 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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[1]

A. Ghose Choudhury, Partha Guha. Chiellini integrability condition, planar isochronous systems and Hamiltonian structures of Liénard equation. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2465-2478. doi: 10.3934/dcdsb.2017126
[2]

Jaume Llibre, Claudia Valls. Analytic integrability of a class of planar polynomial differential systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2657-2661. doi: 10.3934/dcdsb.2015.20.2657
[3]

Antonio Algaba, Cristóbal García, Jaume Giné. Analytic integrability for some degenerate planar systems. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2797-2809. doi: 10.3934/cpaa.2013.12.2797
[4]

Antonio Algaba, María Díaz, Cristóbal García, Jaume Giné. Analytic integrability around a nilpotent singularity: The non-generic case. Communications on Pure and Applied Analysis, 2020, 19 (1) : 407-423. doi: 10.3934/cpaa.2020021
[5]

Fangfang Jiang, Junping Shi, Qing-guo Wang, Jitao Sun. On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047
[6]

Jitsuro Sugie, Tadayuki Hara. Existence and non-existence of homoclinic trajectories of the Liénard system. Discrete and Continuous Dynamical Systems, 1996, 2 (2) : 237-254. doi: 10.3934/dcds.1996.2.237
[7]

Min Hu, Tao Li, Xingwu Chen. Bi-center problem and Hopf cyclicity of a Cubic Liénard system. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 401-414. doi: 10.3934/dcdsb.2019187
[8]

Andrew N. W. Hone, Michael V. Irle. On the non-integrability of the Popowicz peakon system. Conference Publications, 2009, 2009 (Special) : 359-366. doi: 10.3934/proc.2009.2009.359
[9]

Mats Gyllenberg, Yan Ping. The generalized Liénard systems. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 1043-1057. doi: 10.3934/dcds.2002.8.1043
[10]

Na Li, Maoan Han, Valery G. Romanovski. Cyclicity of some Liénard Systems. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2127-2150. doi: 10.3934/cpaa.2015.14.2127
[11]

Shaoyun Shi, Wenlei Li. Non-integrability of generalized Yang-Mills Hamiltonian system. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1645-1655. doi: 10.3934/dcds.2013.33.1645
[12]

Brigita Ferčec, Valery G. Romanovski, Yilei Tang, Ling Zhang. Integrability and bifurcation of a three-dimensional circuit differential system. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4573-4588. doi: 10.3934/dcdsb.2021243
[13]

Hong Li. Bifurcation of limit cycles from a Li$ \acute{E} $nard system with asymmetric figure eight-loop case. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022033
[14]

Leonardo Câmara, Bruno Scárdua. On the integrability of holomorphic vector fields. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 481-493. doi: 10.3934/dcds.2009.25.481
[15]

Andy Hammerlindl. Integrability and Lyapunov exponents. Journal of Modern Dynamics, 2011, 5 (1) : 107-122. doi: 10.3934/jmd.2011.5.107
[16]

Klas Modin, Olivier Verdier. Integrability of nonholonomically coupled oscillators. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 1121-1130. doi: 10.3934/dcds.2014.34.1121
[17]

Bin Liu. Quasiperiodic solutions of semilinear Liénard equations. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 137-160. doi: 10.3934/dcds.2005.12.137
[18]

Robert Roussarie. Putting a boundary to the space of Liénard equations. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 441-448. doi: 10.3934/dcds.2007.17.441
[19]

Sang-hyun Kim, Thomas Koberda. Integrability of moduli and regularity of denjoy counterexamples. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 6061-6088. doi: 10.3934/dcds.2020259
[20]

Menita Carozza, Gioconda Moscariello, Antonia Passarelli. Higher integrability for minimizers of anisotropic functionals. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 43-55. doi: 10.3934/dcdsb.2009.11.43

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (170)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy