A theoretical study of the hysteresis phenomenon at vocal fold oscillation onset-offset

This paper presents a theoretical study on the differences in the biomechanical parameters of the vocal folds between oscillation onset and offset. The dynamics of the oscillation is analyzed from the perspective of the theory of nonlinear dynamical systems, using a mucosal wave model of the vocal folds with the subglottal pressure and the vocal fold half-width as control parameters. It is shown that the oscillation onset occurs through a Hopf bifurcation of the subcritical type, at which an unstable limit cycle is generated. Also, the oscillation offset occurs at a cyclic fold bifurcation, at which the unstable limit cycle and a stable limit cycle (the actual vocal fold oscillation) coalesce and cancel each other. Both bifurcations combine to form an "oscillation hysteresis" phenomenon, common in cases of flow-induced oscillations. An analytical expression for the onset/offset ratio of parameters is derived. The onset/offset ratio is in the range of 0.5-1, in agreement with the experimental evidence. This value depends on the phase delay in motion of the upper edge of the vocal folds versus the lower edge, and on the particular model adopted for airflow separation within the glottis.