  Reed instruments are modeled as self-sustained oscillators driven by the
pressure inside the mouth of the musician. A set of nonlinear equations
connects the control parameters (mouth pressure, lip force) to the system
output, hereby considered as the mouthpiece pressure. Clarinets can then be
studied as dynamical systems, their steady behavior being dictated uniquely by
the values of the control parameters. Considering the resonator as a lossless
straight cylinder is a dramatic yet common simplification that allows for
simulations using nonlinear iterative maps. In this paper, we investigate
analytically the effect of a time-varying blowing pressure on the behavior of
this simplified clarinet model. When the control parameter varies, results from
the so-called dynamic bifurcation theory are required to properly analyze the
system. This study highlights the phenomenon of bifurcation delay and defines a
new quantity, the dynamic oscillation threshold. A theoretical estimation of
the dynamic oscillation threshold is proposed and compared with numerical
simulations.
