Dynamical studies of equations from the Gambier family

Partha Guha; Anindya Ghose Choudhury; Basil Grammaticos

doi:10.3842/SIGMA.2011.028

Dynamical studies of equations from the Gambier family

Partha Guha, Anindya Ghose Choudhury, Basil Grammaticos

Mathematics

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

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.

Original language	British English
Article number	028
Journal	Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Volume	7
DOIs	https://doi.org/10.3842/SIGMA.2011.028
State	Published - 2011

Keywords

Folding transformation
Gambier equation
Milney-pinney equation
Riccati sequence of differential equations
Superintegrability

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10.3842/SIGMA.2011.028

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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{\'e}nard type equation. Finally the conjugate Hamiltonian aspects of certain equations belonging to this family and their connection with superintegrability are explored.",

keywords = "Folding transformation, Gambier equation, Milney-pinney equation, Riccati sequence of differential equations, Superintegrability",

author = "Partha Guha and {Ghose Choudhury}, Anindya and Basil Grammaticos",

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AU - Guha, Partha

AU - Ghose Choudhury, Anindya

AU - Grammaticos, Basil

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AB - 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.

KW - Folding transformation

KW - Gambier equation

KW - Milney-pinney equation

KW - Riccati sequence of differential equations

KW - Superintegrability

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VL - 7

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

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

M1 - 028

ER -

Dynamical studies of equations from the Gambier family

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Keywords

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