Hopf bifurcation in higher dimensional differential systems via the averaging method
type: Item
instance of: scholarly article
title: Hopf bifurcation in higher dimensional differential systems via the averaging method
zbMATH Open document ID: 1171.34027
author: Jaume Llibre
author: Xiang Zhang
published in: Pacific Journal of Mathematics
publication date: 2009-04-08T00:00:00Z
review text: This interesting article treats Hopf bifurcations in higher dimensions. The authors show in their main result that from one singularity with eigenvalues \(\pm bi\) and \(n-2\) zeros, the bifurcation of exactly \(l\) cycles is possible for all \(l\in\{1,\dots, 2^{n-3}\}\). In particular, this shows that the number of limit cycles of a Hopf bifurcation scenario can grow exponentially with the dimension of the system. The result is proved by using first-order averaging theory and Brouwer degree theory. Moreover, the shape and stability of the limit cycles is analyzed in dimension four, and the results are applied to certain fifth-order differential equations. Finally, the authors examine a simplified Marchuk system which models immune response.
reviewed by: Martin Rasmussen
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C29
Mathematics Subject Classification ID: 37G15
zbMATH DE Number: 5541997
zbMATH Keywords: limit cycles
zbMATH Keywords: generalized Hopf bifurcation
zbMATH Keywords: averaging theory
MaRDI profile type: Publication
full work available at URL: pjm.2009.240.321
OpenAlex ID: W2080159910
DOI: 10.2140/PJM.2009.240.321
Recommended article: Hopf bifurcation for some analytic differential systems in \(\mathbb R^3\) via averaging theory
Recommended article: \(3\)-dimensional Hopf bifurcation via averaging theory of second order
Recommended article: 3-dimensional Hopf bifurcation via averaging theory
Recommended article: Averaging approach to cyclicity of Hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems
Recommended article: Q4270699
Recommended article: Q4513246
Recommended article: \(N\)-dimensional zero-Hopf bifurcation of polynomial differential systems via averaging theory of second order
Recommended article: On the 3-dimensional Hopf bifurcation via averaging theory of third order
Recommended article: Hopf Bifurcations of Bi-Parameter Ordinary Differential Systems
Recommended article: HOPF Bifurcation of Singularly Perturbed Autonomous Differential Systems
description: scientific article; zbMATH DE number 5541997