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Automatic identification of dynamical system excited by time-dependent factor without prior information

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Abstract

A moving object is often excited by time-dependent factors. If the equation of motion and time-dependent factor can be distilled from data, a foundation for further analysis can be established. This paper proposes a method to discover the equation of motion and time-dependent factors without prior information and customized preparation. We first construct statistics to determine whether the system is subject to time-dependent factors. For the motivated system, we introduce the time-dependent Fourier series to approximate the excitation. Finally, we achieve the identification by simple steps. The new method does not require the prior knowledge of the system. Since the new method determines whether the time-dependent factor is present and which state variable is motivated, it avoids the confusion caused by inappropriately adding time-dependent terms to the identification. As the designed Fourier series can approximate the unknown model accurately, the new method does not require developing customized identification for each problem. We apply the new method to discover dynamical systems from data collected from Liénard equation and SD oscillator driven by various time-dependent factors. The new method can accurately discover dynamical systems from the collected short-time data and predict many dynamical behaviors, including the long-time evolution of the system, the multi-stable dynamics, and qualitative changes in the dynamics as the time-dependent factor varies. The results show that the new identification method can discover nonautonomous dynamical systems effectively.

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Data Availability Statement

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

References

  1. Hao, R.-B., Lu, Z.-Q., Ding, H., Chen, L.-Q.: A nonlinear vibration isolator supported on a flexible plate: analysis and experiment. Nonlinear Dyn. 108(2), 941–958 (2022). https://doi.org/10.1007/s11071-022-07243-7

    Article  Google Scholar 

  2. Cubitt, T.S., Eisert, J., Wolf, M.M.: Extracting dynamical equations from experimental data is np hard. Phys. Rev. Lett. 108(12), 120503 (2012). https://doi.org/10.1103/PhysRevLett.108.120503

    Article  ADS  CAS  PubMed  Google Scholar 

  3. Afebu, K.O., Liu, Y., Papatheou, E.: Machine learning-based rock characterisation models for rotary-percussive drilling. Nonlinear Dyn. 109(4), 2525–2545 (2022). https://doi.org/10.1007/s11071-022-07565-6

    Article  Google Scholar 

  4. Wu, H.L., Lu, T., Xue, H.Q., Liang, H.: Sparse additive ordinary differential equations for dynamic gene regulatory network modeling. J. Am. Stat. Assoc. 109(506), 700–716 (2014). https://doi.org/10.1080/01621459.2013.859617

    Article  MathSciNet  CAS  PubMed  PubMed Central  Google Scholar 

  5. Li, Y., Xu, S., Duan, J., Liu, X., Chu, Y.: A machine learning method for computing quasi-potential of stochastic dynamical systems. Nonlinear Dyn. 109(3), 1877–1886 (2022). https://doi.org/10.1007/s11071-022-07536-x

    Article  Google Scholar 

  6. Brunton, S.L., Kutz, J.N.: Data-driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press, (2019)

  7. Zhang, Y., Jin, Y., Xu, P., Xiao, S.: Stochastic bifurcations in a nonlinear tri-stable energy harvester under colored noise. Nonlinear Dyn. 99(2), 879–897 (2020). https://doi.org/10.1007/s11071-018-4702-3

    Article  Google Scholar 

  8. Lu, T., Liang, H., Li, H.Z., Wu, H.L.: High-dimensional odes coupled with mixed-effects modeling techniques for dynamic gene regulatory network identification. J. Am. Stat. Assoc. 106(496), 1242–1258 (2011). https://doi.org/10.1198/jasa.2011.ap10194

    Article  MathSciNet  CAS  PubMed  Google Scholar 

  9. Zhang, J., Liu, Y., Zhu, D., Prasad, S., Liu, C.: Simulation and experimental studies of a vibro-impact capsule system driven by an external magnetic field. Nonlinear Dyn. 109(3), 1501–1516 (2022). https://doi.org/10.1007/s11071-022-07539-8

    Article  Google Scholar 

  10. Clemson, P.T., Stefanovska, A.: Discerning non-autonomous dynamics. Phys. Rep. Rev. Sect. Phys. Lett. 542(4), 297–368 (2014). https://doi.org/10.1016/j.physrep.2014.04.001

    Article  MathSciNet  Google Scholar 

  11. Yan, Z., Guirao, J.L.G., Saeed, T., Chen, H., Liu, X.: Analysis of stochastic resonance in coupled oscillator with fractional damping disturbed by polynomial dichotomous noise. Nonlinear Dyn. 110(2), 1233–1251 (2022). https://doi.org/10.1007/s11071-022-07688-w

    Article  Google Scholar 

  12. Ghadami, A., Epureanu, B.I.: Data-driven prediction in dynamical systems: recent developments. Philos. Trans. Royal Soc. Math. Phys. Eng. Sci. 380(2229), 16 (2022). https://doi.org/10.1098/rsta.2021.0213

    Article  Google Scholar 

  13. Bongard, J., Lipson, H.: Automated reverse engineering of nonlinear dynamical systems. Proc. Natl. Acad. Sci. U.S.A. 104(24), 9943–9948 (2007). https://doi.org/10.1073/pnas.0609476104

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  14. Schmidt, M., Lipson, H.: Distilling free-form natural laws from experimental data. Science 324(5923), 81–85 (2009). https://doi.org/10.1126/science.1165893

    Article  ADS  CAS  PubMed  Google Scholar 

  15. Zhang, Y., Duan, J., Jin, Y., Li, Y.: Discovering governing equation from data for multi-stable energy harvester under white noise. Nonlinear Dyn. 106(4), 2829–2840 (2021). https://doi.org/10.1007/s11071-021-06960-9

    Article  Google Scholar 

  16. Huang, N.E., Shen, Z., Long, S.R., Wu, M.L.C., Shih, H.H., Zheng, Q.N., Yen, N.C., Tung, C.C., Liu, H.H.: The empirical mode decomposition and the hilbert spectrum for nonlinear and non-stationary time series analysis. Philos. Trans. Royal Soc. Math. Phys. Eng. Sci. 454(1971), 903–995 (1998). https://doi.org/10.1098/rspa.1998.0193

    Article  MathSciNet  Google Scholar 

  17. Giannakis, D., Majda, A.J.: Nonlinear laplacian spectral analysis for time series with intermittency and low-frequency variability. Proc. Natl. Acad. Sci. U.S.A. 109(7), 2222–2227 (2012). https://doi.org/10.1073/pnas.1118984109

    Article  ADS  MathSciNet  PubMed  PubMed Central  Google Scholar 

  18. Sugihara, G., May, R., Ye, H., Hsieh, C.H., Deyle, E., Fogarty, M., Munch, S.: Detecting causality in complex ecosystems. Science 338(6106), 496–500 (2012). https://doi.org/10.1126/science.1227079

    Article  ADS  CAS  PubMed  Google Scholar 

  19. Kevrekidis, I.G., Gear, C.W., Hyman, J.M., Kevrekidis, P.G., Runborg, O., Theodoropoulos, C.: Equation-free, coarse-grained multiscale computation: enabling microscopic simmulators to perform system-level analysis. Commun. Math. Sci. 1, 715–762 (2003)

    Article  MathSciNet  Google Scholar 

  20. Juang, J.N., Pappa, R.S.: An eigensystem realization-algorithm for modal parameter-identification and model-reduction. J. Guid. Control. Dyn. 8(5), 620–627 (1985). https://doi.org/10.2514/3.20031

    Article  ADS  Google Scholar 

  21. Wu, K.L., Xiu, D.B.: Data-driven deep learning of partial differential equations in modal space. J. Comput. Phys. 408, 22 (2020). https://doi.org/10.1016/j.jcp.2020.109307

    Article  MathSciNet  Google Scholar 

  22. Yoon, R., Bhat, H.S., Osting, B.: A nonautonomous equation discovery method for time signal classification. SIAM J. Appl. Dyn. Syst. 21(1), 33–59 (2022). https://doi.org/10.1137/21m1405216

    Article  MathSciNet  Google Scholar 

  23. Qin, T., Chen, Z., Jakeman, J.D., Xiu, D.B.: Data-driven learning of nonautonomous systems. SIAM J. Sci. Comput. 43(3), 1607–1624 (2021). https://doi.org/10.1137/20m1342859

    Article  MathSciNet  Google Scholar 

  24. Murata, T., Fukami, K., Fukagata, K.: Nonlinear mode decomposition with convolutional neural networks for fluid dynamics. J. Fluid Mech. 882, 15 (2020). https://doi.org/10.1017/jfm.2019.822

    Article  MathSciNet  CAS  Google Scholar 

  25. Mezic, I.: Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41(1–3), 309–325 (2005). https://doi.org/10.1007/s11071-005-2824-x

    Article  MathSciNet  Google Scholar 

  26. Mezic, I.: Analysis of fluid flows via spectral properties of the koopman operator. Annu. Rev. Fluid Mech. 45(45), 357–378 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  27. Rowley, C.W., Mezic, I., Bagheri, S., Schlatter, P., Henningson, D.S.: Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127 (2009). https://doi.org/10.1017/s0022112009992059

    Article  ADS  MathSciNet  Google Scholar 

  28. Schmid, P.J.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28 (2010). https://doi.org/10.1017/s0022112010001217

    Article  ADS  MathSciNet  CAS  Google Scholar 

  29. Mardt, A., Pasquali, L., Wu, H., Noe, F.: Vampnets for deep learning of molecular kinetics. Nat. Commun. 9(1), 5 (2018). https://doi.org/10.1038/s41467-017-02388-1

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  30. Liang, H., Wu, H.L.: Parameter estimation for differential equation models using a framework of measurement error in regression models. J. Am. Stat. Assoc. 103(484), 1570–1583 (2008). https://doi.org/10.1198/016214508000000797

    Article  MathSciNet  CAS  PubMed  PubMed Central  Google Scholar 

  31. Raissi, M., Perdikaris, P., Karniadakis, G.E.: Machine learning of linear differential equations using gaussian processes. J. Comput. Phys. 348, 683–693 (2017). https://doi.org/10.1016/j.jcp.2017.07.050

    Article  ADS  MathSciNet  Google Scholar 

  32. Wei, S., Yan, X., Li, X., Ding, H., Chen, L.-Q.: Parametric vibration of a nonlinearly supported pipe conveying pulsating fluid. Nonlinear Dyn. 111(18), 16643–16661 (2023). https://doi.org/10.1007/s11071-023-08761-8

    Article  Google Scholar 

  33. Brunton, S.L., Proctor, J.L., Kutz, J.N.: Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl. Acad. Sci. U.S.A. 113(15), 3932–3937 (2016). https://doi.org/10.1073/pnas.1517384113

    Article  ADS  MathSciNet  CAS  PubMed  PubMed Central  Google Scholar 

  34. Tibshirani, R.: Regression shrinkage and selection via the lasso. J. Royal Stat. Soc. Series B-Methodol. 58(1), 267–288 (1996). https://doi.org/10.1111/j.2517-6161.1996.tb02080.x

    Article  MathSciNet  Google Scholar 

  35. Fukami, K., Murata, T., Zhang, K., Fukagata, K.: Sparse identification of nonlinear dynamics with low-dimensionalized flow representations. J. Fluid Mech. 926, 35 (2021). https://doi.org/10.1017/jfm.2021.697

    Article  MathSciNet  CAS  Google Scholar 

  36. Champion, K., Lusch, B., Kutz, J.N., Brunton, S.L.: Data-driven discovery of coordinates and governing equations. Proc. Natl. Acad. Sci. U.S.A. 116(45), 22445–22451 (2019). https://doi.org/10.1073/pnas.1906995116

    Article  ADS  MathSciNet  CAS  PubMed  PubMed Central  Google Scholar 

  37. Babaee, H., Sapsis, T.P.: A minimization principle for the description of modes associated with finite-time instabilities. Proceedings of the Royal Society a-Mathematical Physical and Engineering Sciences 472(2186), 27 (2016). https://doi.org/10.1098/rspa.2015.0779

    Article  MathSciNet  Google Scholar 

  38. Bai, Z., Kaiser, E., Proctor, J.L., Kutz, J.N., Brunton, S.L.: Dynamic mode decomposition for compressive system identification. AIAA J. 58(2), 561–574 (2020). https://doi.org/10.2514/1.J057870

    Article  ADS  CAS  Google Scholar 

  39. Wang, B., Wang, L., Peng, J., Hong, M., Xu, W.: The identification of piecewise non-linear dynamical system without understanding the mechanism. Chaos 33(6), 063110 (2023)

    Article  ADS  MathSciNet  PubMed  Google Scholar 

  40. Chartrand, R.: Numerical differentiation of noisy, nonsmooth data. ISRN Appl. Math. 2011, 1–11 (2011). https://doi.org/10.5402/2011/164564

    Article  MathSciNet  Google Scholar 

  41. Stein, E.M., Shakarchi, R.: Fourier Analysis an Introduction. Princeton University Press, (2003)

  42. Eckhoff, K.S.: Accurate reconstructions of functions of finite regularity from truncated fourier-series expansions. Math. Comput. 64(210), 671–690 (1995). https://doi.org/10.2307/2153445

    Article  ADS  MathSciNet  Google Scholar 

  43. Afifi, A., May, S., Clark, V.: Practical multivariate analysis. CRC Press (2011). https://doi.org/10.1201/9781466503243

  44. Han, Y.W., Cao, Q.J., Ji, J.C.: Nonlinear dynamics of a smooth and discontinuous oscillator with multiple stability. Int. J. Bifurcation Chaos 25(13), 16 (2015). https://doi.org/10.1142/s0218127415300384

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work is supported by National Natural Science Foundation of China [grant number 11972289], Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University [grant number CX2022070], and Qin Chuang Yuan “Scientist + Engineer” team construction project in Shaanxi Province [grant number 2023KXJ-268].

Funding

National Natural Science Foundation of China [grant number 11972289], Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University [grant number CX2022070], Qin Chuang Yuan “Scientist + Engineer” team construction project in Shaanxi Province [grant number 2023KXJ-268]

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  1. Department of Applied Probability and Statistics, Northwestern Polytechnical University, Youyi Road 127, Xi’an, 710129, Shaanxi, China

    Wang Bochen, Wang Liang, Peng Jiahui, Dong Shuangqi & Xu Wei

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Correspondence to Wang Liang.

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Bochen, W., Liang, W., Jiahui, P. et al. Automatic identification of dynamical system excited by time-dependent factor without prior information. Nonlinear Dyn 112, 3441–3452 (2024). https://doi.org/10.1007/s11071-023-09232-w

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