Chiellini integrability condition, planar isochronous systems and Hamiltonian structures of Liénard equation

A. Ghose Choudhury; Partha Guha

doi:10.3934/dcdsb.2017126

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2017, Volume 22, Issue 6: 2465-2478. Doi: 10.3934/dcdsb.2017126

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Chiellini integrability condition, planar isochronous systems and Hamiltonian structures of Liénard equation

A. Ghose Choudhury^1, , and
Partha Guha^2,

1.
Department of Physics, Surendranath College, 24/2 Mahatma Gandhi Road, Kolkata -700009, India
2.
S.N. Bose National Centre for Basic Sciences, JD Block, Sector Ⅲ, Salt Lake, Kolkata -700098, India

Author Bio: E-mail address: partha@bose.res.in
A. Ghose Choudhury, E-mail address: aghosechoudhury@gmail.com
A. Ghose Choudhury, E-mail address: aghosechoudhury@gmail.com

Received: June 2016

Revised: February 2017

Published: May 2017

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Abstract

Using a novel transformation involving the Jacobi Last Multiplier (JLM) we derive an old integrability criterion due to Chiellini for the Liénard equation. By combining the Chiellini condition for integrability and Jacobi's Last Multiplier the Lagrangian and Hamiltonian of the Liénard equation is derived. We also show that the Kukles equation is the only equation in the Liénard family which satisfies both the Chiellini integrability and the Sabatini criterion for isochronicity conditions. In addition we examine this result by mapping the Liénard equation to a harmonic oscillator equation using tacitly Chiellini's condition. Finally we provide a metriplectic and complex Hamiltonian formulation of the Liénard equation through the use of Chiellini condition for integrability.

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Mathematics Subject Classification: 34C14, 34C20.

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References

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