Bifurcations and distribution of limit cycles which appear from two nests of periodic orbits
type: Item
instance of: scholarly article
title: Bifurcations and distribution of limit cycles which appear from two nests of periodic orbits
zbMATH Open document ID: 1211.34041
author: R. Asheghi
author: Hamid R. Z. Zangeneh
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2010-08-26T00:00:00Z
review text: Some real planar autonomous differential equations \[ x'=P(x,y),\quad y'=Q(x,y) \] can be written as complex differential equations of the form \[ z'=f(z), \] where \(z=x+i\,y\). This paper deals with the equation \[ z'=iz+z^3. \] Thought as a planar system, it has three critical points of center type, given by the origin and of \(z^\pm=\pm \exp(3\pi i/4)\). Hence, it is natural to study the number of limit cycles that remain when we consider the perturbed differential equation \[ z'=iz+z^3+\varepsilon R_n(z,\bar z), \] where \(\varepsilon\) is real and small enough, and \(R_n\) is a polynomial of degree \(n\) coming from a real polynomial perturbation of the planar system. In fact, this problem has already been studied for the period annulus of the origin [see \textit{A. Gasull, W. Li, J. Llibre} and \textit{Z. Zhang}, Pac. J. Math. 202, No.~2, 341--361 (2002; Zbl 1086.34523)]. The paper under review studies the number of limit cycles that appear simultaneously from the period annuli of the points \(z^\pm\) when the polynomial \(R_m\) has some special shape.
reviewed by: Armengol Gasull
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C08
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 37C27
zbMATH DE Number: 5774955
zbMATH Keywords: weak Hilbert sixteenth problem
zbMATH Keywords: bifurcation of limit cycles
MaRDI profile type: Publication
full work available at URL: j.na.2010.06.009
OpenAlex ID: W2095526788
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DOI: 10.1016/J.NA.2010.06.009
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description: scientific article; zbMATH DE number 5774955