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Rate-dependent bifurcation dodging in a thermoacoustic system driven by colored noise

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Abstract

Tipping in multistable systems occurs usually by varying the input slightly, resulting in the output switching to an often unsatisfactory state. This phenomenon is manifested in thermoacoustic systems. The thermoacoustic instability may lead to the disintegration of rocket engines, gas turbines and aeroengines, so it is necessary to design control measures for its suppression. It was speculated that such unwanted instability states may be dodged by changing the bifurcation parameters quickly enough, and compared with the white noise discussed in [1], colored noise with nonzero correlation time is more practical and important to the system. Thus, in this work, based on a fundamental mathematical model of thermoacoustic systems driven by colored noise, the corresponding Fokker–Planck–Kolmogorov equation of the amplitude is derived by using a stochastic averaging method. A transient dynamical behavior is identified through a probability density analysis. We find that both a relatively higher rate of change of parameters and change in the correlation time of the noise are beneficial to dodge thermoacoustic instability, while a relatively large noise intensity is a disadvantageous factor. More interestingly and importantly, power-law relationships between the maximum amplitude and the noise parameters are uncovered, and the probability of successfully dodging a thermoacoustic instability is calculated. These results serve as a guidance for the design of engines and to propose an effective control strategy, which is of great significance to aerospace-related fields.

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Acknowledgements

This paper was supported by the National Natural Science Foundation of China under Grant Nos. 11772255 and 12072264, the National Key Research and Development Program of China (No. 2018AAA0102201), the Fundamental Research Funds for the Central Universities, the Research Funds for Interdisciplinary Subject of Northwestern Polytechnical University, the Shaanxi Project for Distinguished Young Scholars, the Shaanxi Provincial Key R&D Program 2020KW-013 and 2019TD-010 and the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University under Grant No. CX201962.

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Authors and Affiliations

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

    Xiaoyu Zhang, Yong Xu & Qi Liu

  2. MIIT Key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University, Xi’an, 710072, China

    Yong Xu

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

    Jürgen Kurths

  4. Institute for Complex Systems and Mathematical Biology, School of Natural and Computing Sciences, King’s College, University of Aberdeen, Aberdeen, AB24 3UE, United Kingdom

    Celso Grebogi

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  1. Xiaoyu Zhang
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Correspondence to Yong Xu.

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Zhang, X., Xu, Y., Liu, Q. et al. Rate-dependent bifurcation dodging in a thermoacoustic system driven by colored noise. Nonlinear Dyn 104, 2733–2743 (2021). https://doi.org/10.1007/s11071-021-06368-5

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