Hostname: page-component-7dd5485656-tbj44 Total loading time: 0 Render date: 2025-10-31T17:13:17.615Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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