Dynamical Studies of Equations from the Gambier Family

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.

References in corpus (3)

• The Gambier Mapping, Revisited
• The Gambier Mapping
• On a Novel Class of Integrable ODEs Related to the Painlevé Equations

Cited by in corpus (6)

• Generating Finite Dimensional Integrable Nonlinear Dynamical Systems
• Superposition rules for higher-order systems and their applications
• Mixed superposition rules and the Riccati hierarchy
• On Lie systems and Kummer-Schwarz equations
• A Quasi-Lie Schemes Approach to Second-Order Gambier Equations
• Jacobi-Maupertuis metric of Lienard type equations and Jacobi Last Multiplier