Center problem for trigonometric Liénard systems
type: Item
instance of: scholarly article
title: Center problem for trigonometric Liénard systems
zbMATH Open document ID: 1371.34043
author: Armengol Gasull
author: Claudia Valls
author: Jaume Giné
published in: Journal of Differential Equations
publication date: 2017-08-03T00:00:00Z
full work available at URL: http://ddd.uab.cat/record/182524
review text: The second-order differential equation \(\ddot \theta = g(\theta) + f(\theta) \dot \theta\), where \(f\) and \(g\) are trigonometric polynomials with real coefficients, is equivalent to the first order system \[ \dot \theta = y,\quad \dot y = g(\theta) + f(\theta) y.\eqno{(S)} \] Supposing \(f(0) = g(0) = 0\) and \(g'(0) < 0\) the origin is a focus or a center for (S). The authors prove that the origin is a center if and only if either (a) \(f = \alpha g\) for some real constant \(\alpha\) or (b) there exists a real trigonometric polynomial \(p\) and two real polynomials \(f_1\) and \(g_1\) such that (i) \(p'(0) < 0\), (ii) \(g_1(p(0)) p''(0) < 0\), and (iii) \(f(\theta) = f_1(p(\theta)) p'(\theta)\) and \(g(\theta) = g_1(p(\theta)) p'(\theta)\). The result and techniques used to prove it hark back to the work of L.~A.~Cherkas, C.~J.~Christopher, N.~G.~Lloyd, and J.~M.~Pearson on the characterization of centers in real analytic and polynomial Liénard equations.
reviewed by: Douglas S. Shafer
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 6756304
zbMATH Keywords: center problem
zbMATH Keywords: trigonometric Lienard equation
MaRDI profile type: Publication
OpenAlex ID: W2617504821
cites work: The center-focus problem for a system with homogeneous nonlinearities in the case of zero eigenvalues of the linear part
cites work: Q4123647
cites work: An algebraic approach to the classification of centers in polynomial Liénard systems
cites work: Q4850473
cites work: On general algebraic mechanisms for producing centers in polynomial differential systems
cites work: A simple solution of some composition conjectures for Abel equations
cites work: An explicit bound of the number of vanishing double moments forcing composition
cites work: On the number of limit cycles for perturbed pendulum equations
cites work: Center problem for several differential equations via Cherkas' method
cites work: The number of polynomial solutions of polynomial Riccati equations
cites work: Center conditions for polynomial Liénard systems
cites work: Universal centres and composition conditions
cites work: Limit cycles in the equation of whirling pendulum with autonomous perturbation.
cites work: Limit cycles and chaos in equations of the pendulum type
cites work: Q4379551
cites work: Limit Cycles in the Josephson Equation
cites work: Q5511421
cites work: The classification of reversible cubic systems with center
DOI: 10.1016/J.JDE.2017.05.008
description: scientific article; zbMATH DE number 6756304