The nonlinear dynamics of the vocal fold oscillation at phonation is studied on low dimensional mathematical models. First, a bifurcation diagram is derived for the two-mass model Two equilibrium positions for the vocal folds and associated bifurcation phenomena are found, and a relation with the existence of vocal registers is discussed. It is shown that the results contest a previous oscillation theory derived from a collapsible tube analogy. Next, the phonation threshold pressure is examined on a mucosal wave model. The existence of a hysteresis effect for phonation onset and offset is shown, in agreement with previous experimental results.