Non linear dynamics of the glottis : Phase Portraits and Lyapunov
Succession of the states of a periodic system describes a closed curve in the phase space ... neighboring points in the reference orbit. x0. X1, j x1 x2 x3. X1, j+1.
Non linear dynamics of the glottis : Phase Portraits and Lyapunov Exponents
Maurice Ouaknine
Laboratoire d’Audio-Phonologie Expérimentale et Clinique Université de la Méditerranée Marseille (France)
Signal Analysis Tools available • Wave display • Spectrogram • Jitter • Cepstrum • Autocorrelation • Phase portraits
1
Construction of a Phase Portrait : Pendulum speed
position
speed
time
position
Time signals
Phase space
• Knowledge of the position and the speed of the pendulum is sufficient to characterize its dynamics completely • Succession of the states of a periodic system describes a closed curve in the phase space
Phase Portrait : representation of the dynamics of a system Input
Output
AERODYNAMICS PHYSIOLOGIC
ACTUAL DYNAMICS
SIGNAL
MECANIC
V3
V1,2,3,n : Actual parameters V1
2
Construction of a Phase Portrait : GLOTTIS V3
V2
V1,2,3 : Unknown actual parameters TIME DELAY METHOD D Ruelle, F Takens
V1
Input
Output
AERODYNAMICS PHYSIOLOGIC MECANIC
?
VOCAL SIGNAL
Construction of a Phase Portrait : time delay method
X(t+2τ)
X(t+τ) X(t)
X(t+2τ)
X(t)
τ
X(t+τ)
3
Example 1 : normal voice Sample of sustained Vowel/a/
Quantification of a Phase Portrait • Fractal dimensions – Number of degrees of freedom
• Lyapunov Exponent – Divergence of initially close trajectories • Divergence : Sensitivity to Initial Conditions (SIC)
Determination of Lyapunov exponent
After n iterations, error E E E E n = n n − 1 ... 1 amplification factor would be: E E E E 0 n −1 n − 2 0
Lyapunov coefficient characterizes the logarithm of E increase in relative error during 1 n λ = ∑ log k for iteration n k =1 E k −1
n → ∞ ; E →0 0
Giovanni A, Ouaknine M, Triglia JM. Determination of largest Lyapunov Exponents of Vocal Signal: application to unilateral laryngeal paralysis. J Voice, 1999;13:341-354
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Choice of neighbor points of the reference trajectory
X1, j+2
X1, j+1 1 X1, j
E1
0
x1
E0
E2
X2, j+1
X2, j
2
E2’ E3
x2
x0
X3, j
x3
3
E3’
In an experimental series, the main problem is selection of neighboring points in the reference orbit.
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