Limit cycles bifurcating from a non-isolated zero-Hopf equilibrium of three-dimensional differential systems
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Xiao, Dongmei (Shanghai Jiao Tong University(China). Department of Mathematics)

Date:	2014
Abstract:	In this paper we study the limit cycles bifurcating from a nonisolated zero-Hopf equilibrium of a differential system in R3. The unfolding of the vector fields with a non-isolated zero-Hopf equilibrium is a family with at least three parameters. By using the averaging theory of the second order, explicit conditions are given for the existence of one or two limit cycles bifurcating from such a zero-Hopf equilibrium. To our knowledge, this is the first result on bifurcations from a non-isolated zero-Hopf equilibrium. This result is applied to study three-dimensional generalized Lotka-Volterra systems in [3]. The necessary and sufficient conditions for the existence of a non-isolated zero-Hopf equilibrium of this system are given, and it is shown that two limit cycles can be bifurcated from the non-isolated zero-Hopf equilibrium under a general small perturbation of three-dimensional generalized Lotka-Volterra systems.
Grants:	Ministerio de Economía y Competitividad MTM2008-03437 Agència de Gestió d'Ajuts Universitaris i de Recerca 2009/SGR-410
Note:	Agraïments: The second author has been partially supported by the National Natural Science Foundations of China numbers 10831003 & 10925102, and the Program of Shanghai Subject Chief Scientists number 10XD1406200.
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Three-dimensional differential systems ; Non-isolated zero-Hopf equilibrium ; Limit cycles ; Lotka-Volterra system
Published in:	Proceedings of the American Mathematical Society, Vol. 142 Núm. 6 (2014) , p. 2047-2062, ISSN 1088-6826

DOI: 10.1090/S0002-9939-2014-11923-X

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