Direct and inverse Hopf bifurcation in a neutral delay differential equation model of reed conical instrument
* Presenting author
In conical musical instrument, the self-sustained oscillations appear at a blowingpressure threshold. At this threshold, the mathematical model encounters a so-calledHopf bifurcation. The bifurcation can be either direct or inverse which determines thestability of the arising oscillation. It is well known that for cylindrical instruments(clarinet-like instruments), the bifurcation is direct most of the time. In the case ofconical instruments, the nature of the bifurcation has been studied analytically before,for instance using geometrical approximations based on cylinders. It is expected to findan inverse bifurcation in some cases. The proposed study addresses the bifurcation ofan idealized model of reed instrument with a lossless conical resonator and a compliantmouthpiece, using two methods : a direct time-integration algorithm and a Taylorseries based continuation method of periodic solutions, based on the harmonic balancemethod. It allows to draw the whole branch of periodic solutions and to deduce thetype of Hopf bifurcation from a full diagram. Both direct and inverse bifurcations canbe found, depending on the value of the geometrical parameters of the instrument.