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On the C1 non-integrability of differential systems via periodic orbits

Published online by Cambridge University Press:  06 April 2011

JAUME LLIBRE  and
CLÀUDIA VALLS
JAUME LLIBRE
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain email: jllibre@mat.uab.cat
CLÀUDIA VALLS
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais 1049-001, Lisboa, Portugal email: cvalls@math.ist.utl.pt
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Abstract

We go back to the results of Poincaré [Poincare, H (1891) Sur lintegration des equations differentielles du premier ordre et du premier degre I and II, Rendiconti del circolo matematico di Palermo5, 161–191] on the multipliers of a periodic orbit for proving the C1 non-integrability of differential systems. We apply these results to Lorenz, Rossler and Michelson systems, among others.

Keywords

C1 integrabilitydifferential systemsperiodic orbits
Type
Papers
Information
European Journal of Applied Mathematics , Volume 22 , Issue 4 , August 2011 , pp. 381 - 391
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Arnold, V. I. (2006) Forgotten and neglected theories of Poincaré. Russ. Math. Surv. 61 (1), 1–18.CrossRefGoogle Scholar
[2]Buica, A., Francoise, J. P. & Llibre, J. (2007) Periodic solutions of nonlinear periodic differential systems with a small parameter. Commun. Pure Appl. Anal. 6, 103–111.CrossRefGoogle Scholar
[3]Cairó, L. & Hua, D. (1993) Comments on: ‘integrals of motion for the Lorenz system’. J. Math. Phys. 34, 4370–4371.CrossRefGoogle Scholar
[4]Francoise, J. P. (2005) Oscillations em biologie: Analyse qualitative et modèles. In: Collection: Mathématiques et Applications, Vol. 46, Springer Verlag, Berlin.Google Scholar
[5]Giacomini, H. J., Repetto, C. E. & Zandron, O. P. (1991) Integrals of motion for three-dimensional non-Hamiltonian dynamical systems. J. Phys. A: Math. Gen. 24, 4567–4574.CrossRefGoogle Scholar
[6]Goriely, A. (1996) Integrability, partial integrability, and nonintegrability for systems of ordinary differential equations. J. Math. Phys. 37, 1871–1893.CrossRefGoogle Scholar
[7]Gupta, N. (1993) Integrals of motion for the Lorenz system. J. Math. Phys. 34, 801–804.CrossRefGoogle Scholar
[8]Morales-Ruiz, J. J. (1999) Differential Galois Theory and non-integrability of Hamiltonian systems. In: Progress in Math. Vol. 178, Birkhauser, Verlag, Basel, Switzerland.Google Scholar
[9]Kús, M. (1993) Integrals of motion for the Lorenz system. J. Phys. A: Math. Gen. 16, 689–691.CrossRefGoogle Scholar
[10]Kozlov, V. V. (1983) Integrability and non-integrability in Hamiltonian mechanics. Russ. Math. Surv. 38 (1), 1–76.CrossRefGoogle Scholar
[11]Llibre, J., Buzzi, C. A. & Da Silva, P. R. (2007) 3-dimensional hopf bifurcation via averaging theorem. Discrete Contin. Dyn. Syst. 17, 529–540.CrossRefGoogle Scholar
[12]Lorenz, E. N. (1963) Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141.2.0.CO;2>CrossRefGoogle Scholar
[13]Llibre, J. & Valls, C. (2005) Formal and analytic integrability of the Lorenz system. J. Phys. A 38, 2681–2686.CrossRefGoogle Scholar
[14]Llibre, J. & Valls, C. (2007) Formal and analytic integrability of the Rossler system. Int. J. Bifurcation Chaos Appl. Sci. Eng. 17, 3289–3293.CrossRefGoogle Scholar
[15]Llibre, J. & Valls, C. (2010) The Michelson system is neither global analytic, nor Darboux integrable. Physica D 239, 414–419.CrossRefGoogle Scholar
[16]Llibre, J. & Zhang, X. (2002) Invariant algebraic surfaces of the Lorenz systems. J. Math. Phys. 43, 7613–7635.CrossRefGoogle Scholar
[17]Llibre, J. & Zhang, X. (2002) Darboux integrability for the Rossler system. Int. J. Bifurcation Chaos Appl. Sci. Eng. 12, 421–428.CrossRefGoogle Scholar
[18]Llibre, J. & Zhang, X. (2011) On the Hopf-zero bifurcation of the Michelson system. Nonlinear Analysis: Real World Applications 12, 1650–1653.Google Scholar
[19]Michelson, D. (1986) Steady solutions for the Kuramoto–Sivashinsky equation. Physica D 19, 89–111 (Birkhauser, Verlag, Basel, Switzerland, 1999).CrossRefGoogle Scholar
[20]Poincaré, H. (1891) Sur l'intégration des équations différentielles du premier ordre et du premier degré I and II. Rendiconti del circolo matematico di Palermo 5, 161–191.CrossRefGoogle Scholar
[21]Rossler, E. O. (1976) An equation for continuous chaos. Phys. Lett. A 57, 397–398.CrossRefGoogle Scholar
[22]Schwarz, F. (1985) An algorithm for determining polynomial first integrals of autonomous systems of ordinary differential equations. J. Symbol. Comput. 1, 229–233.CrossRefGoogle Scholar
[23]Segur, H. (1982) Soliton and the inverse scattering transform. In: Osborne, A. R. & Malanotte Rizzoli, P. (editors), Topics in Ocean Physics, North-Holland, Amsterdam, the Netherlands, pp. 235–277.Google Scholar
[24]Steeb, W. H. (1982) Continuous symmetries of the Lorenz model and the Rikitake two–disc dynamo system. J. Phys. A: Math. Gen. 15, 389–390.CrossRefGoogle Scholar
[25]Strelcyn, J. M. & Wojciechowski, S. (1988) A method of finding integrals for three-dimensional systems. Phys. Lett. A 133, 207–212.CrossRefGoogle Scholar
[26]Swinnerton-Dyer, P. (2002) The invariant algebraic surfaces of the Lorenz system. Math. Proc. Camb. Phil. Soc. 132, 385–393.CrossRefGoogle Scholar
[27]Zhang, X. (2002) Exponential factors and Darbouxian first integrals of the Lorenz system. J. Math. Phys. 43, 4987–5001.CrossRefGoogle Scholar