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

Document Type : Research Paper

Authors

1 Department of Mathematics, Yazd University, Yazd, Iran

2 Department of Mathematics, Yazd University, Yazd, Iran.

Abstract

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.

Keywords

[1] H. N. Agiza, E. M. ELabbasy, H. El-Metwally and A. A. Elsadany, Chaotic dynamics of a discrete prey-predator model with Holling type II, Nonlinear Anal. Real World Appl. 10 (2009) 116–129.
[2] A. L. Bojorquez, M. Hirabayashi , J. Hermiz and P. Wang, 1-dimensional model for vocal fold vibration analysis beng 221 problem solving session, Group, 10 (2013) 4.
[3] T. H. Chang and J. T. Lin, Optimal signal timing for an oversaturated intersection, Trans. Res. Part B: Methodological, 34 (2000) 471–491.
[4] L. Cveticanin, Review on mathematical and mechanical models of the vocal cord, J. Appl. Math. Art. ID 928591 (2012), 18 pp.
[5] R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley Publishing Company, Inc., 1989.
[6] M. Fatehi Nia and M. H. Akrami, Stability and bifurcation in a stochastic vocal folds model, Comm. Nonlinear Sci. 79 (2019) 104898.
[7] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields, Springer, New York, 1983.
[8] J. J. Jiang, Y. Zhang and C. McGilligan, Chaos in voice, from modeling to measurement, J. Voice, 20 (2006) 1–18.
[9] Y. A. Kuznetsov, Elements of applied bifurcation theory, 112, Springer Science and Business Media, 2013.
[10] S. E. Levinson, Mathematical Models for Speech Technology, John Wiley and Sons Ltd, 2005.
[11] M. Little, P. McSharry, I. Moroz and S. Roberts, A simple nonlinear model of vocal fold dynamics for synthesis and analysis, NOLISP. 5 (2005), 188–203.
[12] X. Liu and D. Xiao, Complex dynamic behaviors of a discrete-time predator-prey system, Chaos Solitons Fractals, 32, (2007), 80–94.
[13] J. C. Lucero, Bifurcations and limit cycles in a model for a vocal fold oscillator, Commun. Math. Sci. 3 (2005) 517–529.
[14] J. D. Markel, A.H. Gray, Linear Prediction of Speech, Springer-Verlag, 1976.
[15] F. R. Marotto, On redifining on snap-back repller, Chaos Sol. Fract. 25 (2005) 25–28.
[16] J. D. Murray, Mathematical Biology I: An Introduction, 17 Interdisciplinary Applied Mathematics, 2002.
[17] T. M. Onerci, Diagnosis in Otorhinolaryngology, Springer, London, UK, 2010
[18] J. C. Sprott, Chaos and Time-Series Analysis, Oxford University Press, 2003.
[19] I. Steinecke and H. Herzel, Bifurcations in an asymmetric vocal-fold model, J. Acoust. Soc. Amer. 97 (1995) 1874–1884.
[20] J. Sundberg, The science of singing voice, Northern Illinois University Press, Dekalb, Illinois, 1987.
[21] I. R. Titze, Principles of Voice Production, Prentice-Hall, Englewood Cliffs, 1994.
[22] R. Zaccarelli, Mathematical Modelling of Sound Production in Birds, Phd Thesis, Humboldt-Universit zu Berlin, 2008.
Volume 12, Issue 2
November 2021
Pages 305-315
  • Receive Date: 24 December 2019
  • Revise Date: 05 May 2020
  • Accept Date: 08 June 2020