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


SIGMA 7 (2011), 028, 15 pages      arXiv:1103.4210      https://doi.org/10.3842/SIGMA.2011.028
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

Abstract
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.

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