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Research Article | Open Access

Bifurcation and chaos in a discrete activator-inhibitor system

Abdul Qadeer Khan1( ), Zarqa Saleem1, Tarek Fawzi Ibrahim2,3, Khalid Osman4, Fatima Mushyih Alshehri5, Mohamed Abd El-Moneam5
Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad 13100, Pakistan
Department of Mathematics, Faculty of Sciences and Arts (Mahayel), King Khalid University Abha, Saudi Arabia
Department of Mathematics, Faculty of Science, Mansoura University, Egypt
Department of Mathematics, Faculty of Sciences and Arts in Sarat Abeda, King Khalid University, Abha, Saudi Arabia
Department of Mathematics, Faculty of Science, Jazan University, Saudi Arabia
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Abstract

In this paper, we explore local dynamic characteristics, bifurcations and control in the discrete activator-inhibitor system. More specifically, it is proved that discrete-time activator-inhibitor system has an interior equilibrium solution. Then, by using linear stability theory, local dynamics with different topological classifications for the interior equilibrium solution are investigated. It is investigated that for the interior equilibrium solution, discrete activator-inhibitor system undergoes Neimark-Sacker and flip bifurcations. Further chaos control is studied by the feedback control method. Finally, numerical simulations are presented to validate the obtained theoretical results.

Keywords

activator-inhibitor modelNeimark-Sacker bifurcationsflip bifurcationnumerical simulationchaos
CLC number: 70K50, 40A05

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AIMS Mathematics
Volume 8 Issue 2,
February 2023
Pages 4551-4574
DOI: 10.3934/math.2023225

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Cite this article:
Khan AQ, Saleem Z, Ibrahim TF, et al. Bifurcation and chaos in a discrete activator-inhibitor system. AIMS Mathematics, 2023, 8(2): 4551-4574. https://doi.org/10.3934/math.2023225

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Received: 28 July 2022
Revised: 19 October 2022
Accepted: 19 October 2022
Published: 15 February 2023
©2023 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0)