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Volume 5, Issue 3
Zero-Hopf Bifurcation at the Origin and Infinity for a Class of Generalized Lorenz System

Hongpu Liu, Wentao Huang & Qinlong Wang

DOI: 10.12150/jnma.2023.621

J. Nonl. Mod. Anal., 5 (2023), pp. 621-636.

Published online: 2023-08

[An open-access article; the PDF is free to any online user.]

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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 Poincaré 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.

  • Keywords

Generalized Lorenz system, zero-Hopf bifurcation, averaging theory, normal form theory, Poincaré compactification.

  • AMS Subject Headings

34C05, 37C07

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JNMA-5-621, author = {Liu , HongpuHuang , Wentao and Wang , Qinlong}, title = {Zero-Hopf Bifurcation at the Origin and Infinity for a Class of Generalized Lorenz System}, journal = {Journal of Nonlinear Modeling and Analysis}, year = {2023}, volume = {5}, number = {3}, pages = {621--636}, 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 Poincaré 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.

}, issn = {2562-2862}, doi = {https://doi.org/10.12150/jnma.2023.621}, url = {http://global-sci.org/intro/article_detail/jnma/21955.html} }
TY - JOUR T1 - Zero-Hopf Bifurcation at the Origin and Infinity for a Class of Generalized Lorenz System AU - Liu , Hongpu AU - Huang , Wentao AU - Wang , Qinlong JO - Journal of Nonlinear Modeling and Analysis VL - 3 SP - 621 EP - 636 PY - 2023 DA - 2023/08 SN - 5 DO - http://doi.org/10.12150/jnma.2023.621 UR - https://global-sci.org/intro/article_detail/jnma/21955.html KW - Generalized Lorenz system, zero-Hopf bifurcation, averaging theory, normal form theory, Poincaré compactification. AB -

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 Poincaré 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.

Hongpu Liu, Wentao Huang & Qinlong Wang. (2023). Zero-Hopf Bifurcation at the Origin and Infinity for a Class of Generalized Lorenz System. Journal of Nonlinear Modeling and Analysis. 5 (3). 621-636. doi:10.12150/jnma.2023.621
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