Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 028, 15 pages      arXiv:1103.4210
Contribution to the Proceedings of the Conference “Integrable Systems and Geometry”

Dynamical Studies of Equations from the Gambier Family

Partha Guha a, Anindya Ghose Choudhury b and Basil Grammaticos c
a) S.N. Bose National Centre for Basic Sciences, JD Block, Sector-3, Salt Lake, Calcutta-700098, India
b) Department of Physics, Surendranath College, 24/2 Mahatma Gandhi Road, Calcutta-700009, India
c) IMNC, Université Paris VII-Paris XI, CNRS, UMR 8165, Bāt. 104, 91406 Orsay, France

Received December 10, 2010, in final form March 17, 2011; Published online March 22, 2011

We consider the hierarchy of higher-order Riccati equations and establish their connection with the Gambier equation. Moreover we investigate the relation of equations of the Gambier family to other nonlinear differential systems. In particular we explore their connection to the generalized Ermakov-Pinney and Milne-Pinney equations. In addition we investigate the consequence of introducing Okamoto's folding transformation which maps the reduced Gambier equation to a Liénard type equation. Finally the conjugate Hamiltonian aspects of certain equations belonging to this family and their connection with superintegrability are explored.

Key words: Gambier equation; Riccati sequence of differential equations; Milney-Pinney equation; folding transformation; superintegrability.

pdf (354 kb)   tex (17 kb)


  1. Cariñena J.F., Guha P., Rañada M., A geometric approach to higher-order Riccati chain: Darboux polynomials and constants of the motion, J. Phys. Conf. Ser. 175 (2009), 012009, 15 pages.
  2. Cariñena J.F., Guha P., Rañada M., Symplectic structure, Darboux functions and higher-order time-dependent Riccati equations, unpublished.
  3. Torres del Castillo G.F., The Hamiltonian description of a second-order ODE, J. Phys. A: Math. Theor. 43 (2009), 265202, 9 pages.
  4. Duarte L.G.S., Moreira I.C., Santos F.C., Linearization under non-point transformations, J. Phys. A: Math. Gen. 27 (1994), L739-L743.
  5. Ermakov V.P., Second-order differential equations. Conditions of complete integrability, Univ. Isz. Kiev Series III 20 (1880), no. 9, 1-25 (translation by A.O. Harin).
  6. Fris J., Mandrosov V., Smorodinsky Ya.A., Uhlir M., Winternitz P., On higher symmetries in quantum mechanics, Phys. Lett. 16 (1965), 354-356.
  7. Fokas A.S., Yang D., On a novel class of integrable ODEs related to the Painlevé equations, arXiv:1009.5125.
  8. Gambier B., Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est a points critique fixes, Acta Math. 33 (1910), 1-55.
  9. Ghose Choudhury A., Guha P., On isochronous cases of the Cherkas system and Jacobi's last multiplier, J. Phys. A: Math. Theor. 43 (2010), 125202, 12 pages.
  10. Grammaticos B., Ramani A., The Gambier mapping, Phys. A 223 (1996), 125-136, solv-int/9510010.
  11. Grammaticos B., Ramani A., Lafortune S., The Gambier mapping, revisted, Phys. A 253 (1998), 260-270, solv-int/9804011.
  12. Guha P., Stabilizer orbit of Virasoro action and integrable systems, Int. J. Geom. Methods Mod. Phys. 2 (2005), 1-12.
  13. Ince E.L., Ordinary differential equations, Dover Publications, New York, 1944.
  14. Jacobi C.G.J., Sul principio dell'ultimo moltiplicatore, e suo uso come nuovo principio generale di meccanica, Giornale Arcadico di Scienze, Lettere ed Arti 99 (1844), 129-146.
  15. Makarov A.A., Smorodinsky Ya.A., Valiev Kh., Winternitz P., A systematic search for non-relativistic system with dynamical symmetries, Nuovo Cim. A 52 (1967), 1061-1084.
  16. Milne W.E., The numerical determination of characteristic numbers, Phys. Rev. 35 (1930), 863-867.
  17. Painlevé P., Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme, Acta Math. 25 (1902), 1-85.
  18. Pinney E., The nonlinear differential equation y''+p(x)y+cy−3=0, Proc. Amer. Math. Soc. 1 (1950), 681-681.
  19. Ramani A., Grammaticos B., Tamizhmani T., Quadratic relations in continuous and discrete Painlevé equations, J. Phys. A: Math. Gen. 33 (2000), 3033-3044.
  20. Sugai I., Riccati's nonlinear differential equation, Am. Math. Monthly 67 (1960), 134-139.
  21. Sundman K.F., Mémoire sur le problèm des trois corps, Acta Math. 36 (1912), 105-179.
  22. Tsuda T., Okamoto K., Sakai H., Folding transformations of the Painlevé equations, Math. Ann. 331 (2005), 713-738.
  23. Yang D., On conjugate Hamiltonian systems. I. The finite dimensional case, unpublished.

Previous article   Next article   Contents of Volume 7 (2011)