On the stability, chaos and bifurcation analysis of a discrete-time chemostat model using the piecewise constant argument method

Ibraheem M. Alsulami; Ibraheem M. Alsulami

doi:10.3934/math.20241615

AIMS Mathematics

2024, Volume 9, Issue 12: 33861-33878. doi: 10.3934/math.20241615

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On the stability, chaos and bifurcation analysis of a discrete-time chemostat model using the piecewise constant argument method

Ibraheem M. Alsulami ^,

Mathematics Department, Faculty of Science, Umm Al-Qura University, Makkah 21955, Saudi Arabia

Received: 09 September 2024 Revised: 04 November 2024 Accepted: 21 November 2024 Published: 29 November 2024
MSC : 37N25, 39A28, 92D40

In this paper, the dynamics of a discrete-time chemostat model were investigated. The discretization was obtained using the piecewise constant argument method. An analysis was performed to determine the existence and stability of fixed points. In addition, we have shown that the model experiences transcritical and period-doubling bifurcations. Two chaos control techniques, feedback control and hybrid control, were employed to control bifurcation and chaos in the model. Moreover, we provided numerical simulations to substantiate our theoretical results. This study illustrates that the piecewise constant argument method is more dynamically consistent than the forward Euler method.
- chemostat model,
- piecewise constant argument method,
- local stability,
- transcritical bifurcation,
- period-doubling bifurcation,
- chaos control
Citation: Ibraheem M. Alsulami. On the stability, chaos and bifurcation analysis of a discrete-time chemostat model using the piecewise constant argument method[J]. AIMS Mathematics, 2024, 9(12): 33861-33878. doi: 10.3934/math.20241615

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Abstract

In this paper, the dynamics of a discrete-time chemostat model were investigated. The discretization was obtained using the piecewise constant argument method. An analysis was performed to determine the existence and stability of fixed points. In addition, we have shown that the model experiences transcritical and period-doubling bifurcations. Two chaos control techniques, feedback control and hybrid control, were employed to control bifurcation and chaos in the model. Moreover, we provided numerical simulations to substantiate our theoretical results. This study illustrates that the piecewise constant argument method is more dynamically consistent than the forward Euler method.

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