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Occurrence of mixed-mode oscillations in a system consisting of a Van der Pol system and a Duffing oscillator with two potential wells

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Abstract

Mixed-mode oscillations (abbreviated as MMOs) belong to a typical kind of fast/slow dynamical behavior, and how to investigate the mechanism is an important problem in nonlinear dynamics. In this paper, we explore the MMOs induced by the bifurcation delay phenomenon and twist of the trajectories in space based on a coupled system consisting of a Van der Pol system and a Duffing oscillator with two potential wells. Regarding the low-frequency external excitation as a generalized state variable, we obtain the traditional fast and slow subsystems. Appling the equilibrium analysis and bifurcation theory, the stability critical conditions of the equilibrium and the generation conditions of fold and Hopf bifurcation are also presented. To analyze the critical conditions clearly, the two-parameter bifurcation and one-parameter bifurcation diagrams are performed by using numerical simulation method. The bifurcation characteristics are studied, especially the effects of parameter δ on the bifurcation structures. We find that the fast subsystem performs different dynamical behaviors such as fold bifurcation of limit cycles, period-doubling bifurcations, inverse-period-doubling bifurcations and chaos, when parameter δ is taken at different values. By using phase diagrams, time series, maximum Lyapunov exponent diagrams, three-dimensional phase diagrams and superimposed diagrams, the mechanisms of the MMOs are investigated numerically in detail. The Hopf bifurcation delay can lead the trajectories to arrive at the vector fields of the equilibrium point and limit cycles. In addition, the chaotic behaviors can be found on the route of period doubling, which lead to the chaotic spiking-state-oscillations types. Our findings are helpful to understand the generation of the MMOs and intensify the understanding of some special dynamical behaviors on the MMOs.

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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

This paper is supported by the National Natural Science Foundation of China (Grant No. 12002299) and Natural Science Foundation for colleges and universities in Jiangsu Province (Grant No. 20KJB110010).

Funding

Funding for this study was obtained from the National Natural Science Foundation of China (Grant No. 12002299) and the Natural Science Foundation for Colleges and Universities in Jiangsu Province (Grant No. 20KJB110010).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Yancheng Teachers University, Yancheng, 224002, China

    Weipeng Lyu, Shaolong Li & Juanjuan Huang

  2. Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang, 212013, China

    Qinsheng Bi

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  1. Weipeng Lyu
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Lyu, W., Li, S., Huang, J. et al. Occurrence of mixed-mode oscillations in a system consisting of a Van der Pol system and a Duffing oscillator with two potential wells. Nonlinear Dyn 112, 5997–6013 (2024). https://doi.org/10.1007/s11071-024-09322-3

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