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

TL;DR

This paper analytically predicts the delayed onset of sound in a simplified stochastic reed instrument model as the blowing pressure crosses a bifurcation point, accounting for noise effects and validating results with numerical simulations.

Contribution

It introduces an analytical approach to predict delayed bifurcation points in a stochastic reed instrument model, incorporating noise effects and validating with numerical simulations.

Findings

Analytical expressions for bifurcation points with and without noise.

Good agreement between theoretical predictions and numerical simulations.

Identification of stochastic and deterministic dynamic bifurcation points.

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{\^o} 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.