Phonation, or the vibration of the vocal folds, is the primary source of vocalization in the production of voiced sounds by humans. It is a complex bio-mechanical process that is highly sensitive to changes in the speaker's respiratory parameters. Since most symptomatic cases of COVID-19 present with moderate to severe impairment of respiratory functions, we hypothesize that signatures of COVID-19 may be observable by examining the vibrations of the vocal folds. Our goal is to validate this hypothesis, and to quantitatively characterize the changes observed to enable the detection of COVID-19 from voice. For this, we use a dynamical system model for the oscillation of the vocal folds, and solve it using our recently developed ADLES algorithm to yield vocal fold oscillation patterns directly from recorded speech. Experimental results on a clinically curated dataset of COVID-19 positive and negative subjects reveal characteristic patterns of vocal fold oscillations that are correlated with COVID-19. We show that these are prominent and discriminative enough that even simple classifiers such as logistic regression yields high detection accuracies using just the recordings of isolated extended vowels.