Liénard systems and potential-Hamiltonian decomposition. I: Methodology
type: Item
instance of: scholarly article
title: Liénard systems and potential-Hamiltonian decomposition. I: Methodology
zbMATH Open document ID: 1111.37011
author: Jacques Demongeot
author: Nicolas Glade
author: Loïc Forest
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2007-03-02T00:00:00Z
review text: A system of ordinary differential equations on the plane, \(\dot x = f(x,y)\), \(\dot y = g(x,y)\), where \(f\) and \(g\) are polynomials, is called potential-Hamiltonian decomposable (PH-decomposable) if there exist polynomial functions \(P, H : \mathbb{R}^2 \to \mathbb{R}\) such that \(f = -P_x + H_y\) and \(g = -P_y - H_x\), so that the correspodning vector field decomposes as the sum of a gradient and a Hamiltonian vector field, \(-\text{grad}\,P + JdH\). The authors obtain an explicit PH-decomposition for any polynomial system. They indicate its usefulness in studying limit cycles that arise in Liénard systems. [See also the following parts: part II: ibid., No. 3, 191--194 (2007 ; Zbl 1111.37010) and part III: ibid., No. 4, 253--258 (2007; Zbl 1111.37012)].
reviewed by: Douglas S. Shafer
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 5130343
zbMATH Keywords: potential-Hamiltonian decomposition
zbMATH Keywords: planar vector fields
zbMATH Keywords: limit cycles
zbMATH Keywords: polynomial system
MaRDI profile type: Publication
full work available at URL: j.crma.2006.10.016
OpenAlex ID: W2007805301
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DOI: 10.1016/J.CRMA.2006.10.016
description: scientific article; zbMATH DE number 5130343