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Effects of amplitude modulation on mixed-mode oscillations in the forced van der Pol equation

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Abstract

Mixed-mode dynamics is a complex type of dynamical behavior in the classic van der Pol equation with slow forcings. The present paper aims to report the effects of amplitude modulation on mixed-mode dynamics. Typically, interesting dynamical characteristics can be observed in the mixed-mode oscillations (MMOs) when the amplitude modulation is introduced. That is, the quasi-static processes of MMOs exhibit distinct oscillations, while the active phases of MMOs remain largely the same. We investigate these dynamical characteristics by frequency conversion fast–slow analysis. It is found that they are related to the evolution patterns of the amplitude-modulated forcing. In particular, the vibration frequency of the quasi-static processes of MMOs is decided by the modulation frequency, and the transition of MMOs by the modulation amplitude. Our results enrich the routes to fast–slow dynamics and deepen the understanding of dynamical mechanisms of MMOs. Our analysis and treatment provide an important reference for exploring mixed-mode dynamics induced by amplitude modulation in other nonlinear systems.

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Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The authors express their gratitude to the anonymous reviewers whose comments and suggestions have helped improve this paper.

Funding

This work is supported by the National Natural Science Foundation of China (Grant Nos. 12072132 and 12272150).

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Authors and Affiliations

  1. Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang, 212013, People’s Republic of China

    Xiujing Han & Qinsheng Bi

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  1. Xiujing Han
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  2. Qinsheng Bi
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Contributions

XH contributed to conceptualization, methodology, numerical simulations, writing—original draft, supervision, and project administration. QB performed writing—review and editing, and validation.

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Correspondence to Xiujing Han.

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Han, X., Bi, Q. Effects of amplitude modulation on mixed-mode oscillations in the forced van der Pol equation. Nonlinear Dyn 111, 12921–12930 (2023). https://doi.org/10.1007/s11071-023-08517-4

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