Bifurcation and chaos in a discrete activator-inhibitor system

Abdul Qadeer Khan; Zarqa Saleem; Tarek Fawzi Ibrahim; Khalid Osman; Fatima Mushyih Alshehri; Mohamed Abd El-Moneam; Abdul Qadeer Khan; Zarqa Saleem; Tarek Fawzi Ibrahim; Khalid Osman; Fatima Mushyih Alshehri; Mohamed Abd El-Moneam

doi:10.3934/math.2023225

AIMS Mathematics

2023, Volume 8, Issue 2: 4551-4574. doi: 10.3934/math.2023225

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Bifurcation and chaos in a discrete activator-inhibitor system

1.
Department of Mathematics, University of Azad Jammu and Kashmir, Muzaffarabad 13100, Pakistan
2.
Department of Mathematics, Faculty of Sciences and Arts (Mahayel), King Khalid University Abha, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Mansoura University, Egypt
4.
Department of Mathematics, Faculty of Sciences and Arts in Sarat Abeda, King Khalid University, Abha, Saudi Arabia
5.
Department of Mathematics, Faculty of Science, Jazan University, Saudi Arabia

Received: 28 July 2022 Revised: 19 October 2022 Accepted: 19 October 2022 Published: 06 December 2022
MSC : 70K50, 40A05

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.
- activator-inhibitor model,
- Neimark-Sacker bifurcations,
- flip bifurcation,
- numerical simulation,
- chaos
Citation: Abdul Qadeer Khan, Zarqa Saleem, Tarek Fawzi Ibrahim, Khalid Osman, Fatima Mushyih Alshehri, Mohamed Abd El-Moneam. Bifurcation and chaos in a discrete activator-inhibitor system[J]. AIMS Mathematics, 2023, 8(2): 4551-4574. doi: 10.3934/math.2023225

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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.

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