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

Llibre, Jaume; Xiao, Dongmei

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

Limit cycles bifurcating from a non-isolated zero-Hopf equilibrium of three-dimensional differential systems
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by Jaume Llibre and Dongmei Xiao

Proc. Amer. Math. Soc. 142 (2014), 2047-2062

DOI: https://doi.org/10.1090/S0002-9939-2014-11923-X

Published electronically: March 12, 2014

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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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