Dynamical Studies of Equations from the Gambier Family

TL;DR

This paper explores the relationships between Gambier family equations, higher-order Riccati equations, and other nonlinear systems, revealing their connections to integrability, transformations, and Hamiltonian structures.

Contribution

It establishes new links between Gambier equations and various nonlinear differential systems, including Riccati, Ermakov-Pinney, and Lie9nard equations, and examines their Hamiltonian properties.

Findings

Connected Gambier equations to Riccati and Ermakov-Pinney systems.

Applied Okamoto's folding transformation to relate Gambier and Lie9nard equations.

Explored superintegrability and Hamiltonian structures of Gambier family equations.

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.