\(3\)-dimensional Hopf bifurcation via averaging theory of second order
type: Item
instance of: scholarly article
title: \(3\)-dimensional Hopf bifurcation via averaging theory of second order
zbMATH Open document ID: 1186.37059
author: Jaume Llibre
author: Amar Makhlouf
author: Sabrina Badi
published in: Discrete and Continuous Dynamical Systems
publication date: 2009-12-16T00:00:00Z
review text: The authors consider the system \[ \begin{aligned}\dot{x}&=(a_{1} \varepsilon + a_{2} \varepsilon^{2})x - by +f_{1}(x,y,z) + \varepsilon g_{1}(x,y,z) \\ \dot{y} &= bx + (a_{1} \varepsilon + a_{2} \varepsilon^{2} )y + f_{2}(x,y,z) + \varepsilon g_{2}(x,y,z) \\ \dot{z} &= (c_{1} \varepsilon + c_{2} \varepsilon^{2}) z + f_{3}(x,y,x)+ \varepsilon g_{3}(x,y,z) ,\end{aligned} \] where \(f_{i}\) and \(g_{i}\) are quadratic polynomial functions in \(x, y, z\). The method of second order averaging is used to show that at most three limit cycles can bifurcate from the origin when \(\varepsilon = 0\). An example of a system with three limit cycles bifurcated from the origin is also given.
reviewed by: M. A. M. Alwash
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 37G10
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C29
zbMATH DE Number: 5648560
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: limit cycles
zbMATH Keywords: averaging theory
MaRDI profile type: Publication
full work available at URL: dcds.2009.25.1287
OpenAlex ID: W2025865630
DOI: 10.3934/DCDS.2009.25.1287
description: scientific article; zbMATH DE number 5648560