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Stability and Hopf bifurcation analysis of a fractional-order p53 multiple time delays model under PD\(^\alpha \) control

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Abstract

This paper focuses on the analysis of multiple types of time delays and fractional-order proportional-derivative (PD\(^\alpha \)) controller on the dynamics of fractional-order p53 model. The multiple types of time delay schemes are dexterously designed during the bifurcation control for the proposed system, and the comparative study is elaborately performed on bifurcation control theoretically. By analyzing the corresponding characteristic equation, some explicit conditions for the local asymptotical stability of the trivial equilibrium can be given for the fractional-order controlled system. Analysis reveals that the negative feedback delay restrains the delay threshold, while the positive feedback delay promotes the delay threshold and makes the system more robust. System dynamics is affected by fractional order, which is more conducive to the generation of system oscillations. We have adapted control system techniques and designed a PD\(^\alpha \) controller, which is used to activate p53 protein. Numerical results display that the addition of PD\(^\alpha \) controller can increase the richness of the dynamic system. It is worth noting that the differential gain parameter \(k_d\) of PD\(^\alpha \) controller has a significant impact on the system period, and the proportional gain parameter \(k_p\) can amplify the system amplitude. In addition, \(k_p\) can significantly advance the time threshold of system compared to \(k_d\). The PD controller is a good strategy to control the oscillation behaviors of the system.

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Authors and Affiliations

  1. School of Mathematical Sciences, Inner Mongolia University, Hohhot, 010021, China

    Danni Wang, Hongli Yang & Liangui Yang

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  1. Danni Wang
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  2. Hongli Yang
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Correspondence to Hongli Yang.

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Wang, D., Yang, H. & Yang, L. Stability and Hopf bifurcation analysis of a fractional-order p53 multiple time delays model under PD\(^\alpha \) control. Nonlinear Dyn 112, 5663–5686 (2024). https://doi.org/10.1007/s11071-024-09330-3

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