Limit cycles of piecewise differential equations on the cylinder
type: Item
instance of: scholarly article
title: Limit cycles of piecewise differential equations on the cylinder
zbMATH Open document ID: 1486.34074
author: Ali Bakhshalizadeh
author: Jaume Llibre
published in: Bulletin des Sciences Mathématiques
publication date: 2021-07-08T00:00:00Z
review text: The authors consider a class of scalar non-autonomous differential equations with a discontinuity in the independent variable \[ \frac{{dx}}{{dt}} = \begin{cases} f(t,x)& \text{for} \quad 0 \le t \le \pi,\\ g(t,x) & \text{for} \quad \pi \le t \le 2\pi \end{cases} \tag{1} \] with \[ f(t,x) = \sum_{i=0}^n a_i(t)x^i, \qquad g(t,x) = \sum_{j=0}^m b_j(t)x^j, \] where \(a_i\) and \(b_j\) are \(2\pi\)-periodic functions. Thus, (1) can be understood as a differential equation on a cylinder. The maximum number of solutions \(x(t)\) of (1) satisfying \(x(0)=x(2\pi)\) is denoted by \(H(n,m)\). The authors prove \[ H(1,1) = 1, \quad H(1,2) = 2, \quad H(1,3) = \infty. \] Reviewer's remark: The authors forgot to assume some smoothness for \(a_i\) and \(b_j\).
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34A36
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C40
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 7369585
zbMATH Keywords: limit cycles
zbMATH Keywords: piecewise smooth system
zbMATH Keywords: Hilbert number
zbMATH Keywords: differential equations on the cylinder
reviewed by: Klaus R. Schneider
MaRDI profile type: Publication
full work available at URL: j.bulsci.2021.103013
OpenAlex ID: W3171034141
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Wikidata QID: Q115359941
DOI: 10.1016/J.BULSCI.2021.103013
description: scientific article; zbMATH DE number 7369585