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Dynamics of a new modified self-sustained biological trirythmic system with fractional time-delay feedback under correlated noise

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Abstract

The dynamics of a new modified Van der Pol (VDP) self-sustained oscillator, driven by fractional time-delay feedback under correlated noise, is addressed in this paper. The studied system presents a tristability mode with the coexistence of three stable limit cycles in the deterministic case. Under the generalized harmonic balance technique, the fractional derivative simultaneously includes an equivalent quasi-linear dissipative force and quasi-linear restoring force, which reduces the whole problem to an equivalent VDP equation without a fractional derivative. The stochastic averaging method investigates analytical solutions for the equivalent stochastic equation. The critical parametric conditions for stochastic \({\mathcal {P}}\)-bifurcation of amplitude are obtained via the singularity theory for the system switch among the three steady states. The analytical solutions are confronted with direct numerical simulations, in a process where the dynamical features of the system are characterized using the stationary probability density function (PDF) of amplitude and joint PDF of displacement and velocity. A satisfactory agreement is obtained between both approaches, therefore confirming the accuracy of the theoretical predictions. Changing the fractional order, the fractional coefficient, the time delay parameter, and the correlation time also appears to induce the occurrence of the stochastic \({\mathcal {P}}\)-bifurcation.

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Data Availability

The simulation data related to the current study are not publicly available due to but can be obtained from the corresponding author, CBT, on reasonable request.

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Funding

CBT thanks the Kavli Institute for Theoretical Physics (KITP), University of California Santa Barbara (USA), where this work was supported in part by the National Science Foundation Grant no. NSF PHY-1748958, NIH Grant no. R25GM067110, and the Gordon and Betty Moore Foundation Grant no. 2919.01.

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Authors and Affiliations

  1. Fundamental Physics Laboratory, Faculty of Science, University of Bamenda, P.O. Box 39, Bambili, NWR, Cameroon

    R. Mbakob Yonkeu

  2. Laboratory of Biophysics, Department of Physics, Faculty of Science, University of Yaoundé I, P.O. Box 812, Yaoundé, Cameroon

    B. A. Guimfack

  3. Department of Physics and Astronomy, Botswana International University of Science and Technology, Private Bag 16, Palapye, Botswana

    C. B. Tabi & T. C. Kofané

  4. National Advanced School of Engineering of Maroua, University of Maroua, P.O. Box 46, Maroua, Cameroon

    A. Mohamadou

  5. Laboratory of Mechanics, Department of Physics, Faculty of Science, University of Yaounde I, Yaoundé, P.O. Box 812, Cameroon

    T. C. Kofané

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  1. R. Mbakob Yonkeu
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Yonkeu, R.M., Guimfack, B.A., Tabi, C.B. et al. Dynamics of a new modified self-sustained biological trirythmic system with fractional time-delay feedback under correlated noise. Nonlinear Dyn 111, 3743–3764 (2023). https://doi.org/10.1007/s11071-022-07983-6

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