Biomedical Engineering Reference
In-Depth Information
Fig. 9.3.
Locking organization. (
a
)
Approximant of the rotation number
r
ap
as a function of the normalized av-
erage time interval between male notes
T
male
,for
n
= 11 hornero duets. The
bottom axis has been normalized for
each male to the average time inter-
val between notes of the corresponding
female when locked with
r
ap
=1
/
3.
Not every couple displayed the com-
plete locking sequence from 1
/
1to1
/
5,
but all of them displayed 1
/
3atleast
once. (
b
) Rotation number
r
as a func-
tion of forcing period
T
forcing
, for a
nonlinear oscillator subjected to a pe-
riodic forcing. This step-like organiza-
tion is known as the “devil's staircase”.
The bottom axis is normalized to the
natural period of the driven oscillator.
(
c
) Surrogate random duets. The step-
like structure is lost, which is reflected
in the increased overlapping between
steps. This can be quantified by defin-
ing the
average overlap φ
between con-
secutive steps (see text): the original
duets have
φ
∼
20%, while the sur-
rogate duets have
φ
∼
70%. Overlaps
between steps are shown in grey
(a)
1/5
1/4
2/7
3/10
1/3
2/5
1/2
2/3
3/4
1/1
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
T
male
(b)
1/5
2/9
1/4
2/7
1/3
2/5
1/2
2/3
1/1
0.2
0.3
0.4
0.5
0.6
0.7
T
forcing
(c)
1/5
1/4
2/7
1/3
2/5
1/2
2/3
3/4
1/1
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
T
male
a nonlinear dissipation constant, all per unit mass of the labia. This equation
was first proposed to model labial oscillations in oscines. For suboscines, the
tracheal syrinx will show important differences, and actually the membrane
tension can be partially correlated with the pressure [Elemans 2004]. At this
point the model is just a very crude tool to emulate notes. We can do this
by driving (9.2) with a periodic forcing of increasing frequency. For the male,
we chose
p
(
t
)and
k
(
t
) to be harmonic functions of time, with a frequency
