Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities
type: Item
instance of: scholarly article
title: Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities
zbMATH Open document ID: 1405.34027
author: Jianfeng Huang
author: Haihua Liang
author: Jaume Llibre
published in: Journal of Differential Equations
publication date: 2018-09-06T00:00:00Z
full work available at URL: http://ddd.uab.cat/record/199335
review text: This paper studies the number of limit cycles of the system \[ \dot x= ax-y+P_n(x,y),\, \dot y=x+ay+Q_n(x,y), \] where \(a\) is a real parameter and \(P_n\) and \(Q_n\) are real homogeneous polynomials of degree \(n,\) under some additional hypotheses. More concretely, if we introduce the following functions: \(f(\theta)=P_n(\cos \theta,\sin\theta)\cos\theta+Q_n(\cos\theta,\sin\theta)\sin\theta,\) \(g(\theta)=Q_n(\cos \theta,\sin\theta)\cos\theta-P_n(\cos\theta,\sin\theta)\sin\theta,\) \(A(\theta)=ag(\theta)-f(\theta)\) and \(B(\theta)=(n-1)(2ag(\theta)-f(\theta))+g'(\theta)\), the main results of the paper are that the planar system has at most one limit cycle if one of the following conditions holds: (i) there exist two constants \(c_1,c_2\) such that \(c_1^2+c_2^2\neq0\) and \(c_1A(\theta)+c_2B(\theta)\geq0\), or (ii) there exist two nonnegative constants \(d_1\) and \(d_2\) such that \((ad_1)^2+d_2^2\neq0\) and \(A(\theta)(d_1ag(\theta)-d_2f(\theta))\leq0.\) The functions \(f\) and \(g\) appear writing the system in polar coordinates \(r\) and \(\theta\), while the functions \(A\) and \(B\) appear because transforming this expression into an Abel differential equation, by using the so-called Cherkas-Liouville transformation, \(\rho=r^{n-1}/(1+g(\theta)r^{n-1}),\) we arrive to \[ {d\rho}/{d\theta}=(n-1)A(\theta)g(\theta)\rho^3-B(\theta)\rho^2+(n-1)a\rho. \] These results are proved by using that the hypotheses provide some useful curves without contact or invariant by the flow together with the construction of suitable Dulac functions, either for the planar system, or for its expression as an Abel differential equation.
reviewed by: Armengol Gasull
Mathematics Subject Classification ID: 34C07
zbMATH DE Number: 6933401
zbMATH Keywords: limit cycles
zbMATH Keywords: non-existence and uniqueness
zbMATH Keywords: polynomial differential systems
zbMATH Keywords: homogeneous nonlinearities
MaRDI profile type: Publication
OpenAlex ID: W2803987287
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DOI: 10.1016/J.JDE.2018.05.019
description: scientific article; zbMATH DE number 6933401