Bifurcation of limit cycles for 3D Lotka-Volterra competitive systems
type: Item
instance of: scholarly article
title: Bifurcation of limit cycles for 3D Lotka-Volterra competitive systems
zbMATH Open document ID: 1223.34073
author: Haotao Wu
author: Qinlong Wang
author: Wentao Huang
published in: Acta Applicandae Mathematicae
publication date: 2011-05-25T00:00:00Z
review text: The paper analyzes the number of limit cycles in a three-dimensional competitive Lotka-Volterra system. The authors refer to earlier results on Hopf bifurcation in such systems, such as obtained by M. L. Zeeman, J. Hofbauer and J. W.-H. So, and moreover, by Q. Wang, Y. Liu, H. Chen. To calculate the focal values, they use a recursion algorithm that avoids the usual center manifold procedure at an equilibrium with a pair of purely imaginary eigenvalues. For concrete calculations, they treat examples with four parameters and apply the computer algebra system \textit{Mathematica} to determine systems whose first two focal values vanish. This implies the existence of perturbed systems with three limit cycles in six cases of Zeeman's classification. In three of these cases, moreover, the Poincaré-Bendixson theorem yields a fourth limit cycle.
reviewed by: Josef Hainzl
Mathematics Subject Classification ID: 34C60
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 5899552
zbMATH Keywords: three-dimensional Lotka-Volterra system
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: limit cycle
zbMATH Keywords: focal value
zbMATH Keywords: center manifold
describes a project that uses: Mathematica
MaRDI profile type: Publication
full work available at URL: s10440-011-9609-7
OpenAlex ID: W2082390472
cites work: Four limit cycles for a three-dimensional competitive Lotka-Volterra system with a heteroclinic cycle
cites work: Three limit cycles for a three-dimensional Lotka-Volterra competitive system with a heteroclinic cycle
cites work: Hopf bifurcations in competitive three-dimensional Lotka–Volterra systems
cites work: Limit cycles for the competitive three dimensional Lotka-Volterra system
cites work: Asymptotic analysis of interactions between highly conducting cylinders
cites work: Two limit cycles in three-dimensional Lotka-Volterra systems
cites work: Automatic search for multiple limit cycles in three-dimensional Lotka-Volterra competitive systems with classes 30 and 31 in Zeeman's classification
cites work: A 3D competitive Lotka-Volterra system with three limit cycles: a falsification of a conjecture by Hofbauer and So
cites work: Hopf bifurcation for a class of three-dimensional nonlinear dynamic systems
cites work: Applications of centre manifold theory
cites work: A cubic system with twelve small amplitude limit cycles
cites work: On some super fault-tolerant Hamiltonian graphs
cites work: Small limit cycles bifurcating from fine focus points in quartic order \(Z_{3}\)-equivariant vector fields
cites work: Uniformly Persistent Systems
DOI: 10.1007/S10440-011-9609-7
description: scientific article; zbMATH DE number 5899552