Small amplitude limit cycles for the polynomial Liénard system
type: Item
instance of: scholarly article
title: Small amplitude limit cycles for the polynomial Liénard system
zbMATH Open document ID: 1160.34026
author: Henryk Żołądek
author: Maciej Borodzik
published in: Journal of Differential Equations
publication date: 2008-11-06T00:00:00Z
review text: This paper gives an estimation for the maximal number of small amplitude limit cycles bifurcating from the singular point \(x=0\) of focus type for the Liénard equation \[ \ddot x + f(x) \dot x + g(x) = 0, \quad g(0)=0, \;\, g'(0) > 0, \tag{1} \] where \(f\) and \(g\) are real polynomials of degree \(m\) and \(n\), respectively. The authors obtain estimates quadratic in \(m, n\) improving the previous bounds. The Liénard equation (1) is naturally related to the vector field \[ \dot x = y - F(x), \quad \dot y = -G'(x), \quad x, y \in {\mathbb R}^1, \tag{2} \] where \(F\) and \(G\) are real polynomials of degree \(m+1\) and \(n+1\) respectively connected with \(f\) and \(g\) via the relations \(F'(x)=f(x)\), \(F(0)=0\) and \(G'(x)=g(x)\). The vector field (2) can be analytically transformed into the orbitally equivalent one: \[ \dot u = y - \Phi(u), \quad \dot y = -2u, \quad \Phi(u) = c_1u + c_2u^2 + c_3u^3 + \cdots. \] Here, the series \(X = \Phi(Y^{\frac{1}{2}}) = c_1Y^{\frac{1}{2}} + c_2 Y + c_3 Y^{\frac{3}{2}} +\cdots\) is the Puiseux expansion at the origin of the algebraic curve \(C: \;X=F(x), \;Y=G(x)\) on the \((X,Y)\)-plane. Using methods from complex algebraic geometry, the authors estimate the number of double points of the curve \(C\) and prove that the maximal number of double points is \(\delta_{\max} = \frac{1}{2} (mn - \text{ gcd} (m+1,n+1) + 1)\). Let \(\widehat H\) be the maximal number of limit cycles of the vector field (2) bifurcating from the origin, and \(H^*\) be the maximal cyclicity of the origin. Let \(\widehat H_{\mathbb C}\) and \(H^*_{\mathbb C}\) be the similar notions in the complex sense. There are simple relations: \(\widehat H \leq H^* \leq H^*_{\mathbb C} = \widehat H_{\mathbb C}\), and the general estimation \(H^*_{\mathbb C} \leq \frac{1}{2} mn\). In the present paper the authors obtain the following general estimates: \[ H^*_{\mathbb C} \leq \delta_{\max}-1, \quad H^*_{\mathbb C} \leq \tfrac{1}{4} (mn + 3m+3n+1) \tag{3} \] for al…
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 37G15
zbMATH DE Number: 5362102
zbMATH Keywords: Liénard equation
zbMATH Keywords: limit cycles
zbMATH Keywords: singular points
zbMATH Keywords: double points of rational affine curves
reviewed by: Alexey O. Remizov
MaRDI profile type: Publication
full work available at URL: j.jde.2008.07.032
OpenAlex ID: W2088587478
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DOI: 10.1016/J.JDE.2008.07.032
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description: scientific article; zbMATH DE number 5362102