On the qualitative behavior of a class of generalized Liénard planar systems
type: Item
instance of: scholarly article
title: On the qualitative behavior of a class of generalized Liénard planar systems
author: Fabio Zanolin
author: Gabriele Villari
published in: Journal of Dynamics and Differential Equations
publication date: 2022-03-17T00:00:00Z
review text: This work is devoted to the problem of existence of limit cycles for a class of Liénard generalized differential systems \[ \frac{dx}{dt} = y - F(x,y), \quad \frac{dy}{dt} = -g(x), \] assuming that \(F : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}\) and \(g : \mathbb{R} \rightarrow \mathbb{R}\) are locally Lipschitz continuous functions, in order to guarantee the uniqueness of the solutions for the associated initial value problems. It is also assumed that \(g(0) = 0\), \(g(x)x > 0 \) for \(x = 0\) and that the origin is the only singular point of the system. At first, the authors discuss some basic facts of the mentioned system related to the use of the energy of the associated Duffing equations as a Lyapunov function. Then they study the case \(F(x, y) = \lambda B(y)A(x)\), where \(A(x)\) satisfies the standard assumptions on \(F(x)\) in the classical case.
reviewed by: Alexander Grin
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34B30
zbMATH DE Number: 7491607
zbMATH Keywords: generalized Liénard equations
zbMATH Keywords: limit cycles
zbMATH Keywords: qualitative theory of planar dynamical systems
MaRDI profile type: Publication
full work available at URL: s10884-021-09984-2
OpenAlex ID: W3137440577
Wikidata QID: Q115383214
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DOI: 10.1007/S10884-021-09984-2
description: scientific article; zbMATH DE number 7491607