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Energy determines multiple stability in time-delayed systems

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Abstract

Infinite dimensions always challenge the analysis of multiple stability in nonlinear time-delayed systems, as the computation and visualization of conventional basin of attraction are hampered by the increase in systems’ dimensions. To address this issue, this paper introduces an orthonormal basis to approximate the delayed states, uses their signal energy to represent them, and generalises the concept of basin of attraction into stochastic, where each pixel of the basin has the same energy level for each delayed state but corresponds to many initial conditions. Thus, the probabilities are estimated by Monte Carlo method, which is then extensively boosted by artificial neural networks including both classification and regression types. This procedure has been successively applied in the analysis of multiple stability in three typical time-delayed systems, which are a two-dimensional autonomous cutting process, a three-dimensional autonomous neural system, and a two-dimensional non-autonomous forced vibration isolator. They, respectively, have one, two, and two delayed states, with two, three, and five coexisting attractors. It is found that the energy distribution in the delayed state determines both the convergence of Monte Carlo simulation and sensitivity of the classification neural network. It is also seen that the performance of classification neural networks decreases with respect to the increase in the number of attractors, but the regression neural networks show a robuster performance. As a result, the stochastic basin of attraction can be accurately and efficiently computed to reveal the multiple stability in various time-delayed systems.

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Acknowledgements

This research is supported by National Natural Science Foundation of China (Grants No. 11872147, 12072068, 11932015, 11502048, and 11772229) and the Fundamental Research Funds for the Central Universities (Grant No. ZYGX2018J078).

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Authors and Affiliations

  1. School of Aeronautics and Astronautics, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Yao Yan & Qing Guo

  2. School of Aerospace Engineering and Applied Mechanics, Tongji University, 1239 Siping Road, Shanghai, China

    Shu Zhang & Jian Xu

  3. School of Mechanical Engineering, Pusan National University, Busan, 609-735, South Korea

    Kyung Chun Kim

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  1. Yao Yan
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  2. Shu Zhang
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  3. Qing Guo
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Correspondence to Jian Xu.

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Yan, Y., Zhang, S., Guo, Q. et al. Energy determines multiple stability in time-delayed systems. Nonlinear Dyn 102, 2399–2416 (2020). https://doi.org/10.1007/s11071-020-06057-9

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