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Modulating bifurcations in a self-sustained birhythmic system by \(\varvec{\alpha }\)-stable Lévy noise and time delay

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Abstract

Birhythmical nature has been widely encountered and has aroused considerable interest in controlling dynamic behaviors of a self-sustained birhythmic system. However, researchers focus a lot on bifurcations induced by idealized Gaussian noises, the Gaussianity of which can be violated in natural phenomena. The birhythmic oscillator is well suited for modeling biological systems, especially for simulating enzymatic reactions. There are many relevant articles, but few examine the effects of non-Gaussian noises on dynamics of birhythmic systems. \(\alpha \)-stable Lévy noise is more appropriate to characterize complicated biological surroundings. Besides, a few investigations simultaneously introduce the Lévy noise and the inevitable time delay. Control of bifurcations in a birhythmic system driven by \(\alpha \)-stable Lévy noise and time delay is numerically studied in this work, with three noise and two delay parameters treated as control parameters. Modulating the stability index and noise intensity can govern dynamics of the system, but regulating the skewness parameter cannot. More abundant bifurcations are available from adjusting the strength of delayed feedback and time delay. These results may be conducive to further exploring bifurcations in the real-world applications.

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Acknowledgements

This work was partially funded by the National Natural Science Foundation of China under Grant No. 11202120 and the Fundamental Research Funds for the Central Universities under No. GK201901008. The authors also would like to express their appreciation to the reviewers for their insightful reading and constructive comments.

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  1. School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710119, People’s Republic of China

    Lijuan Ning & Yuanli Sun

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  1. Lijuan Ning
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Correspondence to Lijuan Ning.

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Ning, L., Sun, Y. Modulating bifurcations in a self-sustained birhythmic system by \(\varvec{\alpha }\)-stable Lévy noise and time delay. Nonlinear Dyn 98, 2339–2347 (2019). https://doi.org/10.1007/s11071-019-05332-8

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