Prediction of the dynamic oscillation threshold in a clarinet model with a linearly increasing blowing pressure : influence of noise

TL;DR

This paper analyzes how noise and numerical precision affect the dynamic oscillation threshold in a simplified clarinet model with increasing blowing pressure, providing a new theoretical prediction aligned with simulations.

Contribution

It introduces a stochastic model to predict the dynamic oscillation threshold considering noise, extending previous deterministic analyses.

Findings

Dynamic thresholds depend on the increase rate of blowing pressure.

The new theoretical expression accurately predicts thresholds in finite-precision simulations.

Thresholds are independent of initial parameter values.

Abstract

This paper presents an analysis of the effects of noise and precision on a simplified model of the clarinet driven by a variable control parameter. When the control parameter is varied the clarinet model undergoes a dynamic bifurcation. A consequence of this is the phenomenon of bifurcation delay: the bifurcation point is shifted from the static oscillation threshold to an higher value called dynamic oscillation threshold. In a previous work [8], the dynamic oscillation threshold is obtained analytically. In the present article, the sensitivity of the dynamic threshold on precision is analyzed as a stochastic variable introduced in the model. A new theoretical expression is given for the dynamic thresholds in presence of the stochastic variable, providing a fair prediction of the thresholds found in finite-precision simulations. These dynamic thresholds are found to depend on the increase rate and are independent on the initial value of the parameter, both in simulations and in theory.