Bifurcation analysis and chaos control in Zhou's dynamical system
ISSN: 0264-4401
Article publication date: 19 January 2022
Issue publication date: 3 May 2022
Abstract
Purpose
The purpose of the study is to obtain explicit formulas to determine the stability of periodic solutions to the new system and study the extent of the stability of those periodic solutions and the direction of bifurcated periodic solutions. More than that, the authors did a numerical simulation to confirm the results that the authors obtained and presented through numerical analysis are the periodic and stable solutions and when the system returns again to the state of out of control.
Design/methodology/approach
The authors studied local bifurcation and verified its occurrence after choosing the delay as a parameter of control in Zhou 2019’s dynamical system with delayed feedback control. The authors investigated the normal form theory and the center manifold theorem.
Findings
The occurrence of local Hopf bifurcations at the Zhou's system is verified. By using the normal form theory and the center manifold theorem, the authors obtain the explicit formulas for determining the stability and direction of bifurcated periodic solutions. The theoretical results obtained and the corresponding numerical simulations showed that the chaos phenomenon in the Zhou's system can be controlled using a method of time-delay auto-synchronization.
Originality/value
As the delay increases further, the numerical simulations show that the periodic solution disappears, and the chaos attractor appears again. The obtained results can also be applied to the control and anti-control of chaos phenomena of system (1). There are still abundant and complex dynamical behaviors, and the topological structure of the new system should be completely and thoroughly investigated and exploited.
Keywords
Acknowledgements
The authors are grateful to the Editor of the Journal and anonymous referees for their valuable comments.
Funding: No funding was received for this article.
Authors' contributions: All authors have read and approved the final manuscript.
Contributions: EL-Khateeb S ALY (E. S. ALY): Bifurcation analysis, chaos control, software, writing, delay feedback control, numerical results and review editing. M. M. El-Dessoky: Bifurcation analysis, chaos control, software and writing. M. T. Yassen: Bifurcation analysis, chaos control, and numerical simulation. E. Saleh: chaos control, writing and Bifurcation analysis. Ahmed Hussein Msmali: Numerical results, chaos control and review editing. M. A. Aiyashi: Software, writing and chaos control.
Competing interests: The authors declare that they have no competing interests.
Citation
Aly, E.S., El-Dessoky, M.M., Yassen, M.T., Saleh, E., Aiyashi, M.A. and Msmali, A.H. (2022), "Bifurcation analysis and chaos control in Zhou's dynamical system", Engineering Computations, Vol. 39 No. 5, pp. 1984-2002. https://doi.org/10.1108/EC-08-2020-0461
Publisher
:Emerald Publishing Limited
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