# American Institute of Mathematical Sciences

August  2017, 22(6): 2465-2478. doi: 10.3934/dcdsb.2017126

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

 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

A. Ghose Choudhury, E-mail address: aghosechoudhury@gmail.com

Received  June 2016 Revised  February 2017 Published  March 2017

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.

Citation: A. Ghose Choudhury, Partha Guha. Chiellini integrability condition, planar isochronous systems and Hamiltonian structures of Liénard equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2465-2478. doi: 10.3934/dcdsb.2017126
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