Biomedical Engineering Reference
In-Depth Information
dx
dt
= y,
(5.4)
dy
dt
c 2 y,
=
k +( b
β ) y
(5.5)
which has a stationary solution defined by the conditions dx/dt = 0 and
dy/dt = 0; this solution is y =0and x = 0. Now, what are the conditions
under which this solution is stable ? As with the stationary case, we can con-
centrate on the behavior of small departures x and y from this equilibrium,
and just concentrate on a linear approximation to the complete problem.
Neglecting the nonlinear terms, we obtain
d
x
dt
=
y ,
(5.6)
d
y
dt
= −k x +( b − β ) y ,
(5.7)
known as the linearized problem. As it is a linear problem, its solutions are
simple to compute. If we write it in a matrix form d/dt =
DF
, the eigen-
values of the matrix
allow us to gain a good insight into the behavior
of perturbations around the stationary solution. For those parameter values
such that the imaginary parts of the eigenvalues are different from zero, the
system will behave in an oscillatory manner. However, depending on whether
the real part is larger or smaller than zero, those oscillations will lead us
either away from the stationary solution or towards it. For this reason, we
call the curve in parameter space such that the real parts of the eigenvalues
of
DF
are zero (while the imaginary parts may be different from zero) the
bifurcation curve. In this problem, this curve is simply b
DF
β =0.
5.2 The Construction of Syllables
5.2.1 Cyclic Gestures
So far, we have concentrated on the mechanisms by which the labial oscil-
lations that occur during the vocalization of a syllable are turned on and
off. We are now ready to focus on the most remarkable feature of a syllable:
the variation of the fundamental frequency. In terms of the mechanisms that
we have described, an upsweep syllable corresponds to the oscillations being
turned on at a frequency lower than that at which they are turned off. Since
the frequency of the oscillations is determined by the labial restitution coef-
ficient, a variation of the frequency will be produced by a variation of this
parameter. It seems counterintuitive that the labial elasticity can be changed;
it is tempting to suppose that its restitution coe cient K has a fixed value,
determined by the nature of the labial tissue. However, this coe cient can
be controlled. The contraction of the siringealis ventralis muscle increases
 
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