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Transitions in a noisy birhythmic vibro-impact oscillator with improved memory damping regime

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Abstract

We study the critical transition problem for a stochastically forced birhythmic vibro-impact (BVI) model with improved memory damping regime (IMDR) effect. This paper then reports a detailed probabilistic description on the random distribution of the oscillator response. Analytical criteria for critical transitions are obtained successively relying on the amplitude probability, the most probable amplitude (MPA), the joint probability, its cross-sectional view, and the contour projection, which can complement each other. Besides, numerical simulations are carried out to check the theoretical evaluation results. Detect the control parameter source inducing the transition between monorhythmicity and birhythmicity, also indicating the tipping interval of stochastic P-bifurcation and the critical evolution of the half-shaped stochastic attractor mode determined by the vibro-impact constraint, which can also be visually displayed by the intermittent behavior of time history responses. The alternation of feedback gains design of IMDR can subvert the utility direction of transitions caused by some control parameters. Shannon entropy measure is added to display a quantitative indication of the reference interval for the parameters triggering bifurcations, so as to lock effectively in the rhythm mode required by engineering application issues.

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Data availability

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

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (12002250), China Postdoctoral Science Foundation (2020M683429), and the Natural Science Foundation of Shaanxi Province (2021JM-349).

Funding

National Natural Science Foundation of China, 12002250, Deli Wang, Postdoctoral Research Foundation of China, 2020M683429, Deli Wang, Natural Science Foundation of Shaanxi Province, 2021JM-349.

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Authors and Affiliations

  1. School of Science, Xi’an University of Architecture and Technology, Xi’an, 710055, China

    Deli Wang

  2. School of Civil Engineering and Transportation, South China University of Technology, Guangzhou, 510641, China

    Haiqing Pei

  3. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710072, China

    Wei Xu

  4. School of Civil Engineering, Xi’an University of Architecture and Technology, Xi’an, 710055, China

    Jitao Yao

  5. Potsdam Institute for Climate Impact Research, 14412, Potsdam, Germany

    Frank Hellmann & Juergen Kurths

  6. Department of Physics, Humboldt University of Berlin, 12489, Berlin, Germany

    Juergen Kurths

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  1. Deli Wang
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  2. Haiqing Pei
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Correspondence to Deli Wang or Haiqing Pei.

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Wang, D., Pei, H., Xu, W. et al. Transitions in a noisy birhythmic vibro-impact oscillator with improved memory damping regime. Nonlinear Dyn 108, 1045–1070 (2022). https://doi.org/10.1007/s11071-022-07261-5

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