Limit cycles of piecewise polynomial differential systems with the discontinuity line \(xy = 0\)
type: Item
instance of: scholarly article
title: Limit cycles of piecewise polynomial differential systems with the discontinuity line \(xy = 0\)
zbMATH Open document ID: 1490.34038
author: Tao Li
author: Jaume Llibre
published in: Communications on Pure and Applied Analysis
publication date: 2021-12-01T00:00:00Z
review text: The authors study the family of perturbed planar systems \[ \dot x= -yF(x,y)+\varepsilon P(x,y,\varepsilon),\quad \dot y=xF(x,y)+\varepsilon Q(x,y,\varepsilon) \] where \(F=H^m, m\ge0,\) \(H(x,y)=(x^2+y^2)/2\) and \(P\) and \(Q\) are piecewise polynomial perturbations, with discontinuities at \(xy=0.\) Notice that the unperturbed system has the first integral \(H\) and a global center at the origin. This fact makes it very suitable to be studied by using the averaging theory. The paper gives upper bounds for the number of limit cycles appearing by using the first non identically null averaged function in terms of the degree of the perturbation and the order of this averaged function. These bounds are sharpened either by assuming that the perturbations are hamiltonian or that these perturbations have some special symmetries.
reviewed by: Armengol Gasull
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34E10
Mathematics Subject Classification ID: 34A36
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 34C29
Mathematics Subject Classification ID: 37J40
zbMATH DE Number: 7436400
zbMATH Keywords: averaging theory
zbMATH Keywords: limit cycles
zbMATH Keywords: bifurcation
zbMATH Keywords: piecewise polynomial differential equations
MaRDI profile type: Publication
full work available at URL: cpaa.2021136
OpenAlex ID: W3188525739
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DOI: 10.3934/CPAA.2021136
description: scientific article; zbMATH DE number 7436400