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On the analytic integrability of the Liénard analytic differential systems

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


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  • [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.


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


    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.


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


    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.


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


    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.


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


    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.


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


    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.


    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.


    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.


    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.


    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.

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