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Limit cycles bifurcating from a non-isolated zero-Hopf equilibrium of three-dimensional differential systems


Authors: Jaume Llibre and Dongmei Xiao
Journal: Proc. Amer. Math. Soc. 142 (2014), 2047-2062
MSC (2010): Primary 37N25, 34C12, 34C28, 37G20
DOI: https://doi.org/10.1090/S0002-9939-2014-11923-X
Published electronically: March 12, 2014
MathSciNet review: 3182024
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Abstract: In this paper we study the limit cycles bifurcating from a non-isolated zero-Hopf equilibrium of a differential system in $ \mathbb{R}^3$. The unfolding of the vector fields with a non-isolated zero-Hopf equilibrium is a family with at least three parameters. By using analysis techniques and 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. This result is applied to study three-dimensional generalized Lotka-Volterra systems in a paper by Bobieński and Żołądek (2005). 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.


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Additional Information

Jaume Llibre
Affiliation: Departament de Matemátiques, Universitat Autónoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
Email: jllibre@mat.uab.cat

Dongmei Xiao
Affiliation: Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
Email: xiaodm@sjtu.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-2014-11923-X
Received by editor(s): November 12, 2011
Received by editor(s) in revised form: July 4, 2012
Published electronically: March 12, 2014
Additional Notes: The first author was supported by the grants MEC/FEDER MTM 2008-03437, CIRIT 2009SGR 410 and ICREA Academia
The second author was supported by the National Natural Science Foundations of China numbers 10831003 and 10925102 and the Program of Shanghai Subject Chief Scientists number 10XD1406200.
Communicated by: Yingfei Yi
Article copyright: © Copyright 2014 American Mathematical Society

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