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Sufficient statistics for parameter inference in the stochastic Van der Pol oscillator model for thermoacoustic instabilities

Y. Sungtaek Ju*

  • *Contact author: sungtaek.ju@ucla.edu
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Phys. Rev. E 114, 014203 – Published 7 July, 2026

DOI: https://doi.org/10.1103/k56x-bf4g

Abstract

Thermoacoustic instabilities pose a critical challenge in the design and operation of many combustion systems. We report a low-dimensional statistics approach for inferring the parameters of the stochastic Van der Pol oscillator from trajectory observations. From the exponential family structure of the stationary amplitude distribution, we identify sufficient statistics that capture essential parametric information. To break the parameter degeneracy inherent to stationary distribution methods, we incorporate the amplitude autocorrelation time, which provides independent information about noise intensity through relaxation dynamics. The inverse mapping from sufficient statistics to parameters can be effectively approximated by quadratic polynomial regression, achieving comparable accuracy to neural networks while requiring orders of magnitude fewer parameters and providing interpretable closed-form expressions. Comparison with previous parameter identification approaches reveals complementary strengths. In the critical regime near the Hopf bifurcation, our method enables regime classification with reasonable accuracy. Under measurement noise, a periodogram-based moment correction yields graceful degradation across all three parameter estimates, with residual high-noise error dominated by autocorrelation-time bias rather than moment bias. The methodology provides interpretable symbolic expressions suitable for thermoacoustic stability monitoring, and can be extended to other stochastic oscillator systems.

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References (44)

  1. T. C. Lieuwen and V. Yang, Combustion Instabilities in Gas Turbine Engines: Operational Experience, Fundamental Mechanisms, and Modeling, Progress in Aeronautics and Astronautics Vol. 210 (AIAA, Reston, VA, 2005).
  2. T. Poinsot, Prediction and control of combustion instabilities in real engines, Proc. Combust. Inst. 36, 1 (2017).
  3. M. P. Juniper and R. I. Sujith, Sensitivity and nonlinearity of thermoacoustic oscillations, Annu. Rev. Fluid Mech. 50, 661 (2018).
  4. N. Noiray and B. Schuermans, Deterministic quantities characterizing noise driven Hopf bifurcations in gas turbine combustors, Int. J. Non Linear Mech. 50, 152 (2013).
  5. N. Noiray, Linear growth rate estimation from dynamics and statistics of acoustic signal envelope in turbulent combustors, J. Eng. Gas Turbines Power 139, 041503 (2017).
  6. B. van der Pol and J. van der Mark, The heartbeat considered as a relaxation oscillation, and an electrical model of the heart, London, Edinburgh, Dublin Philos. Mag. J. Sci. 6, 763 (1928).
  7. K. Grudziński and J. J. Żebrowski, Modeling cardiac pacemakers with relaxation oscillators, Physica A 336, 153 (2004).
  8. R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1, 445 (1961).
  9. J. Nagumo, S. Arimoto, and S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE 50, 2061 (1962).
  10. R. Stoop, T. Jasa, Y. Uwate, and S. Martignoli, From hearing to listening: Design and properties of an actively tunable electronic hearing sensor, Sensors 7, 3287 (2007).
  11. R. Belousov, F. Berger, and A. J. Hudspeth, Volterra-series approach to stochastic nonlinear dynamics: Linear response of the Van der Pol oscillator driven by white noise, Phys. Rev. E 102, 032209 (2020).
  12. J. Venkatramani, S. Sarkar, and S. Gupta, Intermittency in pitch-plunge aeroelastic systems explained through stochastic bifurcations, Nonlinear Dyn. 92, 1225 (2018).
  13. A. P. Kuznetsov, S. P. Kuznetsov, and N. V. Stankevich, A simple autonomous quasiperiodic self-oscillator, Commun. Nonlinear Sci. Numer. Simul. 15, 1676 (2010).
  14. N. Noiray and A. Denisov, A method to identify thermoacoustic growth rates in combustion chambers from dynamic pressure time series, Proc. Combust. Inst. 36, 3843 (2017).
  15. I. V. Sysoev and B. P. Bezruchko, Noise robust approach to reconstruction of van der Pol-like oscillators and its application to Granger causality, Chaos 31, 083118 (2021).
  16. J. Timmer, Parameter estimation in nonlinear stochastic differential equations, Chaos Solit. Fractals 11, 2571 (2000).
  17. E. Boujo and N. Noiray, Robust identification of harmonic oscillator parameters using the adjoint Fokker–Planck equation, Proc. R. Soc. A 473, 20160894 (2017).
  18. S. J. Lade, Finite sampling interval effects in Kramers–Moyal analysis, Phys. Lett. A 373, 3705 (2009).
  19. C. Honisch and R. Friedrich, Estimation of Kramers-Moyal coefficients at low sampling rates, Phys. Rev. E 83, 066701 (2011).
  20. G. Bonciolini, E. Boujo, and N. Noiray, Output-only parameter identification of a colored-noise-driven Van-der-Pol oscillator: Thermoacoustic instabilities as an example, Phys. Rev. E 95, 062217 (2017).
  21. M. Lee, V. Gupta, and L. K. B. Li, System identification of a low-density jet via its noise-induced dynamics, J. Fluid Mech. 862, 200 (2019).
  22. M. Lee, Y. Guan, V. Gupta, and L. K. B. Li, Input-output system identification of a thermoacoustic oscillator near a Hopf bifurcation using only fixed-point data, Phys. Rev. E 101, 013102 (2020).
  23. T. Indlekofer, A. Faure-Beaulieu, J. R. Dawson, and N. Noiray, Dynamics of azimuthal thermoacoustic modes in imperfectly symmetric annular combustors, J. Fluid Mech. 944, A15 (2022).
  24. H. Son and M. Lee, A PINN approach for identifying governing parameters of noisy thermoacoustic systems, J. Fluid Mech. 984, A21 (2024).
  25. M. Xie, X. Zhao, D. Zhao, J. Fu, C. Shelton, and B. Semlitsch, Predicting bifurcation and amplitude death characteristics of thermoacoustic instabilities from PINNs-derived van der Pol oscillators, J. Fluid Mech. 998, A46 (2024).
  26. S. Mariappan, A. Mukhopadhyay, H. Son, and M. Lee, Learning thermoacoustic interactions in combustors using a physics-informed neural network, Eng. Appl. Artif. Intell. 138, 109388 (2024).
  27. T. Sugandi, B. Dharmaputra, and N. Noiray, A deep operator network for Bayesian parameter identification of self-oscillators, Data-Centric Eng. 5, e35 (2024).
  28. A. Nóvoa and L. Magri, Real-time thermoacoustic data assimilation, J. Fluid Mech. 948, A35 (2022).
  29. A. Nóvoa, N. Noiray, J. R. Dawson, and L. Magri, A real-time digital twin of azimuthal thermoacoustic instabilities, J. Fluid Mech. 1001, A49 (2024).
  30. R. L. Stratonovich, Topics in the Theory of Random Noise (Gordon and Breach, New York, 1963), Vol. 1.
  31. R. Z. Khasminskii, On the principle of averaging the Itô stochastic differential equations, Kybernetika 4, 260 (1968).
  32. C. W. Gardiner, Stochastic methods: A Handbook for the Natural and Social Sciences, 4th ed. (Springer, Berlin, 2009).
  33. N. G. van Kampen, Stochastic Processes in Physics and Chemistry, 3rd ed. (Elsevier, North-Holland, Amsterdam, 2007).
  34. H. Risken, The Fokker–Planck Equation: Methods of Solution and Applications, 2nd ed. (Springer, Berlin, Heidelberg, 1996), Vol. 18.
  35. E. L. Lehmann and G. Casella, Theory of Point Estimation, 2nd ed. (Springer, New York, 1998).
  36. S. L. Brunton, J. L. Proctor, and J. N. Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems, Proc. Natl. Acad. Sci. USA 113, 3932 (2016).
  37. L. Boninsegna, F. Nüske, and C. Clementi, Sparse learning of stochastic dynamical equations, J. Chem. Phys. 148, 241723 (2018).
  38. J. L. Callaham, J.-C. Loiseau, G. Rigas, and S. L. Brunton, Nonlinear stochastic modelling with Langevin regression, Proc. R. Soc. A 477, 20210092 (2021).
  39. D. A. Messenger and D. M. Bortz, Weak SINDy: Galerkin-based data-driven model selection, Multiscale Model. Simul. 19, 1474 (2021).
  40. P. Subramanian, R. I. Sujith, and P. Wahi, Subcritical bifurcation and bistability in thermoacoustic systems, J. Fluid Mech. 715, 210 (2013).
  41. V. Nair, G. Thampi, and R. I. Sujith, Intermittency route to thermoacoustic instability in turbulent combustors, J. Fluid Mech. 756, 470 (2014).
  42. L. Magri, M. P. Juniper, and J. P. Moeck, Sensitivity of the Rayleigh criterion in thermoacoustics, J. Fluid Mech. 882, R1 (2020).
  43. A. Krishnan, R. Manikandan, P. Miber, R. I. Sujith, N. Marwan, and J. Kurths, Suppression of thermoacoustic instability by targeting the hubs of the turbulent networks in a bluff-body stabilized combustor, J. Fluid Mech. 916, A20 (2021).
  44. K. Kashinath, L. K. B. Li, and M. P. Juniper, Forced synchronization of periodic and aperiodic thermoacoustic oscillations: Lock-in, bifurcations and open-loop control, J. Fluid Mech. 838, 690 (2018).

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Phys. Rev. E 114, 014203– Published 7 July, 2026
  • Received 7 February 2026
  • Accepted 23 June 2026
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