Upper bounds for the number of limit cycles through linear differential equations
type: Item
instance of: scholarly article
title: Upper bounds for the number of limit cycles through linear differential equations
zbMATH Open document ID: 1135.34027
author: Armengol Gasull
author: Hector Giacomini
published in: Pacific Journal of Mathematics
publication date: 2007-07-20T00:00:00Z
review text: Consider the differential system \[ \dot x= y,\quad\dot y= h_0(x)+ h_1(x) g+ h_2(x) y^2+ y^3\tag{\(*\)} \] in the plane, where the functions \(h_i\), \(i= 0,1,2\), are smooth. To any natural number \(n\geq 1\), the authors associate with \((*)\) two functions \(f_n(x,y)\) and \(M_n(x)\), where \(M_n\) is a solution of a linear differential equation of order \(n+1\) which does not change sign. Under some additional assumptions on \(f_n\) and \(M_n\) it is proved that any limit cycle of \((*)\) does not cut the curve \(f_n(x,y)= 0\) and that the number of limit cycles of \((*)\) in an \(\ell\)-connected component of \(\mathbb{R}^2\setminus\{f_n(x, y)= 0\}\) is at most \(\ell\), and that all these limit cycles are hyperbolic. The proof is based on the Bendixson-Dulac criterion for \(\ell\)-connected sets. Some concrete examples are treated.
reviewed by: Klaus R. Schneider
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 5173442
MaRDI profile type: Publication
full work available at URL: pjm.2006.226.277
OpenAlex ID: W2015610311
DOI: 10.2140/PJM.2006.226.277
Recommended article: Q2954142
Recommended article: Q5409436
Recommended article: Upper bounds on the number of limit cycles in generalized Liénard equations of odd type
Recommended article: The number of limit cycles of certain polynomial differential equations
Recommended article: Number of Limit Cycles for Some Non-generic Classes of Piecewise Linear Differential Systems
Recommended article: Upper bounds for the number of limit cycles of some planar polynomial differential systems
Recommended article: Maximum number of limit cycles for certain piecewise linear dynamical systems
Recommended article: Q4394633
Recommended article: Q4930144
Recommended article: ON THE NUMBER OF LIMIT CYCLES IN PIECEWISE-LINEAR LIÉNARD SYSTEMS
description: scientific article; zbMATH DE number 5173442