Biomedical Engineering Reference
In-Depth Information
Fig. 9.5.
Locking organization for the Poincare oscillator, periodically kicked along
the horizonal axis. The “Arnold tongues” are the regions of the parameter space
(
T,A
) where stable solutions of different rotation numbers can be found, identified
here by different gray levels.
T
= forcing period (in units of the period of the
oscillator
T
0
=2
π/ω
);
A
= forcing amplitude
integers
r
=
p/q
,where
q
represents the period of the stable solution (in units
of the forcing period), and
p
the number of complete cycles that the forced
system performs before repeating itself. In this way, the period-one solutions
of (9.9) can be labeled by
r
=1
/
1 (the white region in Fig. 9.5), while
the period-two solutions of (9.12) can be labeled by
r
=1
/
2 (the first gray
level). For a given value of
A
, we could plot the rotation numbers obtained
in successive numerical simulations performed for different forcing periods.
The result would be a more or less complex stair, like the one described in
Sect. 9.2. But what is the dynamical mechanism behind the existence of these
“steps”?
Figure 9.6 allows us to gain some intuition. In this figure, we show the
(graphical) solution to the fixed points of the second-iterate map at para-
meter values corresponding to the begining and the end of the period-two
tongue, for a constant amplitude
A
=0
.
8. In other words, the intersections
between the straight line and the curve represent those values of
φ
fulfilling
(9.12). Notice that it is possible that there may be no intersections at all;
the existence of intersections is governed by the values of the parameters
T
and
A
, which set the position of the curve in the plane. As the forcing fre-
quency is increased, the main change in the curve that can be described is a
global shift downwards. However, the curve preserves its qualitative shape,
and this is why we have found no intersections for
T<
0
.
47, two intersections
