Calculation of the steady-state oscillations of a flute model using the orthogonal collocation method

In this paper we exploit the method of numerical continuation combined with orthogonal collocation to determine the steady-state oscillations of a model of flute-like instruments that is formulated as a nonlinear neutral delay differential equation. The delay term in the model causes additional complications in the analysis of the behaviour of the model, in contrast to models of other wind instruments which are formulated as ordinary differential equations. Fortunately, numerical continuation provides bifurcation diagrams that show branches of stable and unstable static and periodic solutions of the model and their connections at bifurcations, thus enabling an in depth analysis of the global dynamics. Furthermore, it allows us to predict the thresholds of the different registers of the flute-like instrument and thus to explain the classical phenomenon of register change and the associated hysteresis.