Limit cycle bifurcation from a zero-Hopf equilibrium for a class of 3-dimensional Kolmogorov systems
Bouaziz, Chamseddine (University of Annaba. Department of Mathematics)
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Makhlouf, Ammar (University of Annaba. Department of Mathematics)

Date:	2024
Abstract:	A zero-Hopf equilibrium point p of a 3-dimensional autonomous differential system in R3 is an equilibrium point such that the eigenvalues of the linear part of the system at p are 0 and ±ωi with ω≠0. A zero-Hopf bifurcation takes place when from a zero-Hopf equilibrium bifurcate some small-amplitude limit cycles moving the parameters of the system. Polynomial Kolmogorov systems are the natural extension of the polynomial Lotka-Volterra systems of degree 2 to higher degree. The Kolmogorov systems have been studied intensively due to their applications for modeling many natural phenomena. In this paper we characterize, up to first order in the averaging theory, the eight distinct zero-Hopf bifurcations which can exhibit the class of 3-dimensional Kolmogorov systems of degree 3, providing an explicit approximation of the bifurcated small-amplitude limit cycles, together with information about their kind of stability. From each one of this eight different zero-Hopf bifurcations emerges one or two limit cycle using the averaging theory of first order. Moreover we provide an explicit example of each one of these eight zero-Hopf bifurcations.
Grants:	Agencia Estatal de Investigación PID2022-136613NB-I00 European Commission 777911 Agència de Gestió d'Ajuts Universitaris i de Recerca 2021/SGR-00113
Note:	Altres ajuts: Real Acadèmia de Ciències i Arts de Barcelona
Rights:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, la comunicació pública de l'obra i la creació d'obres derivades, sempre que no sigui amb finalitats comercials, i sempre que es reconegui l'autoria de l'obra original.
Language:	Anglès
Document:	Article ; recerca ; Versió publicada
Subject:	Kolmogorov system ; Periodic orbit ; Averaging theory ; Zero-Hopf bifurcation ; Zero-Hopf equilibria
Published in:	Partial Differential Equations in Applied Mathematics, Vol. 11 (September 2024) , art. 100810, ISSN 2666-8181

DOI: 10.1016/j.padiff.2024.100810

24 p, 1.5 MB

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Research literature > UAB research groups literature > Research Centres and Groups (research output) > Experimental sciences > GSD (Dynamical systems)
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