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Dynamical Studies of Equations from the Gambier Family

By Partha Guha, Anindya Ghose Choudhury and Basil Grammaticos

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 Li\'enard type equation. Finally the conjugate Hamiltonian aspects of certain equations belonging to this family and their connection with superintegrability are explored

Topics: Nonlinear Sciences - Exactly Solvable and Integrable Systems, Mathematics - Classical Analysis and ODEs
Year: 2011
DOI identifier: 10.3842/SIGMA.2011.028
OAI identifier: oai:arXiv.org:1103.4210
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