Small-amplitude limit cycles in polynomial Liénard systems
type: Item
instance of: scholarly article
title: Small-amplitude limit cycles in polynomial Liénard systems
zbMATH Open document ID: 0855.34030
DOI: 10.1007/BF01195913
author: Colin Christopher
author: Noel G. Lloyd
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 1997-02-16T00:00:00Z
review text: For the Liénard equation \(d^2x/dt^2 + f(x) dx/dt + g(x) = 0\) or its equivalent two dimensional system (1) \(dx/dt = y\), \(dy/dt = - g(x) - yf(x)\), assume that \(f\) and \(g\) are polynomials with \(f(0) = g(0) = 0\), \(g'(0) > 0\). The origin is then either a fine focus or a centre. The paper is concerned with the order of such a fine focus. Let \({\mathcal L} (n,m)\) denote the class of Liénard systems in which \(g'(0) > 0\) and degrees of \(f\) and \(g\) are \(n\) and \(m\), respectively, and let \(H(n,m)\) be the maximum order of a fine focus for systems in \({\mathcal L} (n,m)\). Each system (1) in \({\mathcal L} (n,m)\) can be identified by the pair \((f,g)\). The following results are obtained by the authors. Theorem. Let \(\varphi (f,g)\) be the order of the fine focus at the origin for (1), and \({\mathcal L}_0 (n,m) = \{(f,g) \mid (f,g) \in {\mathcal L} (n,m)\) and \(f'(0) > 0\}\), \(H_0 (n,m) = \max \{\varphi (f,g) \mid (f,g) \in {\mathcal L}_0 (n,m)\}\). (i) If \((f,g) \in {\mathcal L}_0 (n,m)\), then \(\varphi (f,g) = \varphi (g,f)\). (ii) \([{1 \over 2} (n + m - 1)] \leq H_0 (n,m) \leq [{1 \over 2} nm]\). (iii) For \(n,m > 1\), \(H_0 (n,m) = H_0 (m,n)\). (iv) For \(n \geq m\), we have \(H(n,m) =H_0 (n,m) \leq H(m,n)\).
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 885324
zbMATH Keywords: Liénard equation
zbMATH Keywords: fine focus
zbMATH Keywords: centre
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 885324