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Bifurcation analysis in the system with the existence of two stable limit cycles and a stable steady state

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Abstract

Van der Pol–Duffing oscillator, which can be used a model for many dynamical system, has been widely concerned. However, most of the systems by scholars are either stable steady states or limit cycles. Here, the self-sustained oscillator with the coexistence of steady state and limit cycles, which is famous for describing the flutter of airfoils with large span ratio in low-speed wind tunnels, is treated in this paper. Using the energy balance method, the deterministic bifurcation of the tristable system with time-delay feedback is investigated. The presence of time-delay feedback expands the bifurcation range of the parameters, making the bifurcation phenomenon more abundant. In addition, according to the stationary probability density function obtained by the stochastic averaging method, stochastic bifurcation of the system with time-delay feedback and noise is explored theoretically. The numerical results confirm the correctness of the theoretical analysis. Transition between the unimodal structure, the bimodal structure and the trimodal structure is found. Many rich bifurcations are available by adjusting the time-delay and noise intensity, which may be conductive to achieve the desired phenomenon in the real-world application.

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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

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Ning, L. Bifurcation analysis in the system with the existence of two stable limit cycles and a stable steady state. Nonlinear Dyn 102, 115–127 (2020). https://doi.org/10.1007/s11071-020-05887-x

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