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Geometry of non-standard Hamiltonian structures of Liénard equations and contact structure

José F. Cariñena

https://orcid.org/0000-0003-4480-6535

Departamento de Física Teórica, Universidad de Zaragoza, 50009 Zaragoza, Spain

E-mail Address: jfc@unizar.es

Corresponding author.

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and

Partha Guha

https://orcid.org/0000-0003-2843-2668

Department of Mathematics, Khalifa University of Science and Technology Main Campus — Zone 1, Abu Dhabi, United Arab Emirates

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA

E-mail Address: partha.guha@ku.ac.ae

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https://doi.org/10.1142/S021988782440005XCited by:4 (Source: Crossref)

This article is part of the issue:

20th Anniversary Special Issue — From Quantum Information to Cosmology: Exploring the Frontiers of Geometric Methods in Physics

Abstract

We start with a self-contained brief review of the construction of non-standard Lagrangian and Hamiltonian structures for the Liénard equations satisfying Chiellini condition, we apply it to a special polynomial class of Liénard equation and explore the integrability structure. Then after a brief exposition of the contact geometry and its connection with the non-standard Hamiltonian structures, we describe the time evolution of the contact Hamiltonian and Liénard equation. We present the formulation of the Liénard equation in terms of General Equation for the Non-Equilibrium Reversible–Irreversible Coupling (GENERIC) method and also study the gradient-type flow. Finally, we describe the generalized Liénard equation by using the conformal Hamiltonian mechanics and illustrate our construction using spatio-temporal and autocatalysis systems.

Keywords:

PACS: 02.30.Hq, 45.20.−d, 45.20.Jj, 05.45.−a, 45.50.Jf

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