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| Zero-Hopf Bifurcation at the Origin and Infinity for a Class of Generalized Lorenz System |
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| KeyWord:Generalized Lorenz system, zero-Hopf bifurcation, averaging theory, normal form theory, Poincar$\acute{e}$ compactification |
| Author Name | Affiliation | | Hongpu Liu | School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, China | | Wentao Huang | School of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, China | | Qinlong Wang | School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, China
Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation\, $\&$\, Center for Applied Mathematics of Guangxi (GUET), Guilin 541002, China |
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| Abstract: |
| In this paper, the zero-Hopf bifurcations are studied for a generalized Lorenz system. Firstly, by using the averaging theory and normal form theory, we provide sufficient conditions for the existence of small amplitude periodic solutions that bifurcate from zero-Hopf equilibria under appropriate parameter perturbations. Secondly, based on the Poincar{\'e} compactification, the dynamic behavior of the generalized Lorenz system at infinity is described, and the zero-Hopf bifurcation at infinity is investigated. Additionally, for the above theoretical results, some related illustrations are given by means of the numerical simulation. |
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