Bifurcation Analysis of a Vocal Fold Model Coupled to Resonators (original) (raw)

Bifurcations and limit cycles in a model for a vocal fold oscillator

Communications in Mathematical Sciences, 2005

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.

Dynamics of the two-mass model of the vocal folds: Equilibria, bifurcations, and oscillation region

The Journal of the Acoustical Society of America, 1993

The dynamics of the large-amplitude oscillation of the vocal folds is analyzed using the two-mass model. First, the equilibrium positions are determined in the case of a rectangular prephonatory glottis, and the existence of two equilibrium positions besides the rest position is shown. Their stability is examined and a bifurcation diagram is derived with a normalized subglottal pressure and a coupling coefficient as control parameters. Phase plane plots are shown to illustrate the results. The cases of convergent and divergent prephonatory glottis are then briefly considered. The main results are finally discussed relative to previous analytical works; it is shown that they disprove the previous oscillation theory based on the existence of a glottal negative differential resistance.

Effect of source–tract acoustical coupling on the oscillation onset of the vocal folds

The Journal of the Acoustical Society of America, 2012

This paper analyzes the interaction between the vocal folds and vocal tract at phonation onset due to the acoustical coupling between both systems. Data collected from a mechanical replica of the vocal folds show that changes in vocal tract length induce fluctuations in the oscillation threshold values of both subglottal pressure and frequency. Frequency jumps and maxima of the threshold pressure occur when the oscillation frequency is slightly above a vocal tract resonance. Both the downstream and upstream vocal tracts may produce those same effects. A simple mathematical model is next proposed, based on a lumped description of tissue mechanics, quasi-steady flow and one-dimensional acoustics. The model shows that the frequency jumps are produced by saddle-node bifurcations between limit cycles forming a classical pattern of a cusp catastrophe. The transition from a low frequency oscillation to a high frequency one may be achieved through two different paths: in case of a large aco...

Bifurcations and chaos in register transitions of excised larynx experiments

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2008

Experimental data from an excised larynx are analyzed in the light of nonlinear dynamics. The excised larynx provides an experimental framework that enables artificial control and direct observation of the vocal fold vibrations. Of particular interest in this experiment is the coexistence of two distinct vibration patterns, which closely resemble chest and falsetto registers of the human voice. Abrupt transitions between the two registers are typically accompanied by irregular vibrations. Two approaches are presented for the modeling of the excised larynx experiment; one is the nonlinear predictive modeling of the experimental time series and the other is the biomechanical modeling ͑three-mass model͒ that takes into account basic mechanisms of the vocal fold vibrations. The two approaches show that the chest and falsetto vibrations correspond to two coexisting limit cycles, which jump to each other with a change in the bifurcation parameter. Irregular vibrations observed at the register jumps are due to chaos that exists near the two limit cycles. This provides an alternative mechanism to generate chaotic vibrations in excised larynx experiment, which is different from the conventionally known mechanisms such as strong asymmetry between the left and right vocal folds or excessively high subglottal pressure.

Interaction of vocal fold and vocal tract oscillations

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We study the mechanical feedback coupling between the human vocal folds and vocal tract (VT) by simulating fundamental frequency glides over the lowest VT resonance. In the classical sourcefilter theory of speech production, the vocal folds produce a signal which is filtered by the resonator, vocal tract without any feedback. We have developed a computational model of the vocal folds and the VT that also includes a counter pressure from the VT to the vocal folds. This coupling gives rise to new computational observations (such as modal locking) that can be established experimentally.

Oscillation region of a piecewise-smooth model of the vocal folds

Communications in Mathematical Sciences, 2006

The two-mass model of the vocal folds is a popular representation of their dynamical structure used in phonation studies. This paper presents an analysis of a recent piecewise-smooth version of the model. This version has two equilibrium positions, and in one of them (the initial prephonatory position) the system is nondifferentiable. Standard methods of stability analysis do not apply for that position, because they require smoothness of the system. A geometrical approach is applied instead, which is an extension of a method previously developed for planar systems. The analysis shows the existence of a transcritical bifurcation between the equilibrium positions, and a Hopf bifurcation related to each of them. The oscillation region of the model is next determined as the area delimited by the Hopf bifurcations. The results are illustrated by a bifurcation diagram and trajectory plots.

Chest- and falsetto-like oscillations in a two-mass model of the vocal folds

The Journal of the Acoustical Society of America, 1996

The dynamics of the two-mass model of the vocal fold oscillation is analyzed. It is shown that the oscillation may occur around two equilibrium positions, and that each case presents similar features as the chest and falsetto registers, suggesting a relation between them. The switch between equilibrium positions is caused by a transcritical bifurcation phenomenon.

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

The Journal of the Acoustical Society of America, 1999

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 exp...

Bifurcation and chaos in a one mass discrete time vocal fold dynamical model

International Journal of Nonlinear Analysis and Applications, 2021

In this article, we are going to study the stability and bifurcation of a two-dimensional discrete time vocal fold model. The existence and local stability of the unique fixed point of the model is investigated. It is shown that a Neimark-Sacker bifurcation occurs and an invariant circle will appear. We give sufficient conditions for this system to be chaotic in the sense of Marotto. Numerically it is shown that our model has positive Lyapunov exponent and is sensitive dependence on initial conditions. Some numerical simulations are presented to illustrate our theoretical results.

Continuous model for vocal fold oscillations to study the effect of feedback

Physical Review E, 2001

In this work we study the effects of delayed feedback on vocal fold dynamics. To perform this study, we work with a vocal fold model that is made as simple as possible while retaining the spectral content characteristic of human vocal production. Our results indicate that, even with the simplest explanation for vocal fold oscillation, delayed feedback due to reflected sound in the vocal tract can lead to extremely rich dynamics.