Analytical prediction of delayed Hopf bifurcations in a simplified stochastic model of reed musical instruments

TL;DR

This work tackles the onset of sound in a simplified single-reed instrument when the blowing pressure increases over time in the presence of white-noise forcing. By applying stochastic averaging, the authors derive a slow, one-dimensional Itô SDE for the mouthpiece-pressure amplitude and identify dynamic bifurcation points—deterministic and stochastic—that mark when oscillations reemerge as the control parameter crosses the static Hopf point. They obtain analytical expressions for these dynamic bifurcation values, delineate regimes based on noise, and validate the theory with numerical simulations, showing good agreement in mean-square amplitude growth and in the amplitude PDFs. The results illuminate how noise and rate of pressure increase influence bifurcation delay and sound onset, with implications for playing dynamics and instrument design, and they lay groundwork for extending the approach to more complex reed-instrument models.

Abstract

This paper investigates the dynamic behavior of a simplified single reed instrument model subject to a stochastic forcing of white noise type when one of its bifurcation parameters (the dimensionless blowing pressure) increases linearly over time and crosses the Hopf bifurcation point of its trivial equilibrium position. The stochastic slow dynamics of the model is first obtained by means of the stochastic averaging method. The resulting averaged system reduces to a non-autonomous one-dimensional It{ô} stochastic differential equation governing the time evolution of the mouthpiece pressure amplitude. Under relevant approximations the latter is solved analytically treating separately cases where noise can be ignored and cases where it cannot. From that, two analytical expressions of the bifurcation parameter value for which the mouthpiece pressure amplitude gets its initial value back are deduced. These special values of the bifurcation parameter characterize the effective appearance of sound in the instrument and are called deterministic dynamic bifurcation point if the noise can be neglected and stochastic dynamic bifurcation point otherwise. Finally, for illustration and validation purposes, the analytical results are compared with direct numerical integration of the model in both deterministic and stochastic situations. In each considered case, a good agreement is observed between theoretical results and numerical simulations, which validates the proposed analysis.

Figures (13)

• Figure 1: Main steps of the proposed approach as a block diagram.
• Figure 2: Deterministic dynamic bifurcation point $\hat{y}^{\text{dyn}}_{\text{det}}$, given by Eq. \ref{['eq:dyndetbifpt1']}, as a function of the initial condition $y_0$. The opposite of the static bifurcation point $\hat{\gamma}^{\text{st}}$, given by Eq. \ref{['eq:statbifpt']}, is depicted by a vertical dashed line. The red points correspond to the initial conditions used in Fig. \ref{['fig:pDBPDetNum']}. The set of parameters \ref{['eq:param1']} is used.
• Figure 3: Numerical simulations of the deterministic differential equation associated to Eq. \ref{['eq:VdP1Moy3Stochay']} using a logarithm scale for the vertical axis. The set of parameters \ref{['eq:param1']} is used with, in addition, $x(y_0)=0.01$ (the latter is depicted by a horizontal dashed line) and $y_0=-0.36,-0.32,\dots,-0.16$. The static bifurcation point $\hat{y}^{\text{st}}=0$ is depicted by a vertical gray dashed line. The dynamic bifurcation point $\hat{y}^{\text{dyn}}_{\text{det}}$ is defined for a given initial condition by the value of $y$ for which $|x(y)|=|x(y_0)|$, which corresponds graphically to the intersection between the blue curves and the horizontal dashed line. The green and red points are used respectively to highlight the considered initial conditions and the corresponding dynamic bifurcation points predicted by Eq. \ref{['eq:dyndetbifpt1']}.
• Figure 4: The deterministic bifurcation point $\hat{y}^{\text{dyn}}_{\text{det}}$ given by Eq. \ref{['eq:dyndetbifpt1']} as a function of the initial condition $y_0$ for five values of the damping coefficient, i.e. $\alpha_1=0,0.025\dots,0.1$ with $\omega_1=1000$ rad$\cdot$s$^{-1}$, $F_1=1200$ s$^{-1}$ and $\zeta=0.2$. Red dashed lines indicate the value of $-\hat{\gamma}^{\text{st}}$ for each value of $\alpha_n$.
• Figure 5: The deterministic bifurcation point $\hat{y}^{\text{dyn}}_{\text{det}}$ given by Eq. \ref{['eq:dyndetbifpt1']} as a function of the parameter $\zeta$ for four of the damping coefficient ($\alpha_1=0,0.01,0.02$ and $0.05$), for three value of the initial condition ($y_0=-\hat{\gamma}^\text{st},-3\hat{\gamma}^\text{st}/4$ and $-\hat{\gamma}^\text{st}/2$) with $\omega_1=1000$ rad$\cdot$s$^{-1}$ and $F_1=1200$ s$^{-1}$.

Theorems & Definitions (1)

• Definition 4.1: Dynamic pitchfork bifurcation point