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.
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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

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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/

-Quasi-periodic attractor -Closed-loop trajectories

Example 2 : severe dysphonia

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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.

Signal instability : clinical measurements

jitter

Lyapunov

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