Zero-Hopf Bifurcation at the Origin and Infinity for a Class of Generalized Lorenz System |
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| Authors: | Hongpu Liu Wentao Huang Qinlong Wang |
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| Affiliation: | School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, China;School of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, China; 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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| Keywords: | Generalized Lorenz system zero-Hopf bifurcation averaging theory normal form theory Poincar$acute{e}$ compactification |
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