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Bifurcation analysis in the system with the existence of three stable limit cycles

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Abstract

The purpose of this manuscript is to analyze the bifurcation of  the tri-rhythmic model with three stable limit cycles under the excitation of Gaussian colored noise, which aims to reveal the extremely complex nonlinear dynamics in biology. In the deterministic bifurcation of the tri-rhythmic model with time-delay, an interesting tri-rhythmic collapse and recovery phenomenon is found. By tuning the time-delay feedback, the model also exhibits a switch between tri-rhythmic and bi-rhythmic behavior. Furthermore, the tri-rhythmic model under colored noise excitation is discussed theoretically based on the stationary probability density function obtained by the stochastic averaging method. The variation in the number of peak values of the stationary probability density function is observed, thereupon enabling the stochastic bifurcation of the tri-rhythmic oscillator. Therefore, by adjusting the time-delay feedback and colored noise, many abundant bifurcation phenomena can be acquired. The theoretical analysis is consistent with the results of the dynamic numerical simulation. Our study may provide a new perspective for further exploring the bifurcation phenomenon of the tri-rhythmic model, which has important practical significance.

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Acknowledgements

This work is partially supported by the Natural Science Foundation of Shaanxi Province (Grant No. 2022JM-040).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, China

    Jing Yuan, Lijuan Ning & Qin Guo

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  1. Jing Yuan
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  2. Lijuan Ning
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  3. Qin Guo
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Correspondence to Lijuan Ning.

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Yuan, J., Ning, L. & Guo, Q. Bifurcation analysis in the system with the existence of three stable limit cycles. Indian J Phys 98, 1767–1781 (2024). https://doi.org/10.1007/s12648-023-02927-1

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