This article presents an analysis of the dynamics of a bidimensional oscillator, which has been proposed as a simple model for the vocal fold motion at phonation. The model is capable of producing an oscillation with physiologically realistic values for the parameters. A simple extension of the model using even-powered polynomials in the damping factor is proposed, to permit the occurrence of an oscillation hysteresis phenomenon commonly observed in voice onset-offset patterns. This phenomenon appears from the combination of a subcritical Hopf bifurcation where an unstable limit cycle is produced, with a fold bifurcation between limit cycles, where the unstable limit cycle coalesces and cancels with a stable limit cycle. The results are illustrated with phase plane plots and bifurcation diagrams obtained using numerical continuation techniques.
"Bifurcations and limit cycles in a model for a vocal fold oscillator." Commun. Math. Sci. 3 (4) 517 - 529, December 2005.