Input-output system identification of a thermoacoustic oscillator near a Hopf bifurcation using only fixed-point data

We present a framework for performing input-output system identification near a Hopf bifurcation using data from only the fixed-point branch, prior to the Hopf point itself. The framework models the system with a van der Pol–type equation perturbed by additive noise, and identifies the system parameters via the corresponding Fokker-Planck equation. We demonstrate the framework on a prototypical thermoacoustic oscillator (a flame-driven Rijke tube) undergoing a supercritical Hopf bifurcation. We find that the framework can accurately predict the properties of the Hopf bifurcation and the limit cycle beyond it. This study constitutes an experimental demonstration of system identification on a reacting flow using only prebifurcation data, opening up pathways to the development of early warning indicators for nonlinear dynamical systems near a Hopf bifurcation.

Figure 1

Schematic of the experimental setup consisting of a prototypical thermoacoustic system (a flame-driven Rijke tube) perturbed by extrinsic noise from a loudspeaker. DAQ: data acquisition system.

Figure 2

Probability density function of the pressure fluctuation amplitude on the fixed-point branch ($z/L=0.256$) for three different noise amplitudes ($d$).

Figure 3

Identification of the actuator model coefficients, where $n$ is the gradient of subfigure (a) and $k$ is the gradient of subfigure (b). The vertical intercept of subfigure (b) is the background noise amplitude ($b$), which is negligible and thus consistent with our modeling assumptions.

Figure 4

Power spectral density showing a noise-induced peak and its Lorentzian fit on the fixed-point branch ($z/L=0.267$) for $d=4.0\times {10}^{-4}$. The horizontal axis is the normalized frequency ($\widehat{\omega }$), with $2\mathrm{\Delta }\omega$ denoting the width at half maximum of the Lorentzian fit.

Figure 5

(a) Experimental bifurcation diagram of the system, where the horizontal axis is the normalized flame position ($z/L$) measured from the top of the combustor. Also shown is (b) the PSD of the pressure fluctuations as a function of $z/L$.

Figure 6

Experimental probability density function (black dots) and its surface interpolation on the fixed-point branch ($z/L=0.267$), just before the Hopf point. For all the noise amplitudes tested, $P\left(a\right)$ is unimodal, confirming the supercritical nature of the Hopf bifurcation.

Figure 7

Determining the DVDP coefficients via SI. Extrapolation is performed using data from only the fixed-point branch (black diamonds), after the removal of outliers (gray diamonds), which are defined here as being outside three standard deviations. The extrapolation is performed with a linear model for the linear coefficient ($\epsilon$) and with a power-law model for the nonlinear coefficients (${\alpha }_1, {\alpha }_2, {\alpha }_3$). The predicted data (red circles) are on the limit-cycle branch, whose features are examined in Fig. 10.

Figure 8

Comparison of bifurcation diagrams between the experimental system and the numerical model found via SI. The Hopf point predicted by the model is at $z/L=0.269$, which is within the experimentally observed range: $0.267<z/L<0.273$. The blue line represents the experimental data bandpass filtered around the limit-cycle frequency ($f_1\pm 10$ Hz).

Figure 9

Anisochronicity of the experimental system: (a) peak frequency $f_{pk}$ as a function of the noise amplitude $d$ and (b) power spectral density at different values of $d$ on the fixed-point branch ($z/L=0.267$), just before the Hopf point. The frequency shift is observed to be less than $0.3\%$.

Figure 10

Comparison of phase portraits between the experimental system and the numerical model found via SI at four different positions on the limit-cycle branch: $z/L=0.273$ (a), 0.285 (b), 0.297 (c), and 0.308 (d). The experimental data are shown both in unfiltered form (gray) and in bandpass-filtered form (blue: $f_1\pm 10$ Hz), while the numerical data are shown in unfiltered form only (red).