Biomedical Engineering Reference
In-Depth Information
φ
n
+1
=
α
+
φ
a
(
φ
n
,A
)
,
(9.6)
where
φ
a
is the change in phase that occurs when the system is kicked.
The phase
φ
a
depends on the phase
φ
n
of the system at the moment it is
kicked and on the size of the kick
A
. From Fig. 9.4, it is simple to derive
an expression for this new phase. Notice that when the system is displaced
from point 1 to point 2, we can define a triangle with vertices at the origin
and at points 1 and 2. The sides are of lengths
L
1
=
ρ
= 1 (the
value of
the radial coordinate),
L
2
=
A
and
L
3
=
A
2
+1
φ
n
)(bythe
cosine theorem). Projecting the three sides of our triangle onto the
x
axis,
we can write
−
2
A
cos(
π
−
L
3
cos(
φ
a
(
φ
n
,A
)) = cos(
φ
n
)+
A,
(9.7)
which together with (9.6), gives us an expression for the phase of the oscillator
in the
n
th step as a function of the phase in the previous step. This reads
,
A
+cos(
φ
n
)
A
2
+1
α
+cos
−
1
φ
n
+1
=
P
(
φ
n
)
≡
(9.8)
−
2
A
cos(
π
−
φ
n
)
which is known as the phase map of the Poincare oscillator.
9.3.3 Stable Periodic Solutions
Equation (9.8) was easy to find, but it does not look easy to work with.
Nevertheless, it can help us in our process of gaining an understanding of
how complex patterns are generated by such systems. First of all, we have to
learn how to interpret the solutions of these maps in terms of the solutions of
the original problem: a forced oscillator. The simplest temporal pattern that
could emerge out of the forcing of a nonlinear system is a “one-to-one” locked
state: the system repeats its behavior after a time equal to the time between
forcing kicks. How would this pattern be represented in the map formalism?
Since this formalism only inspects what the system is doing at discrete times,
an underlying periodic solution would be captured by a map formalism as a
fixed point of the discrete dynamics. In other words, a period-one solution
(measured in units of the forcing period) corresponds to a situation in which
the phase of the system at a given kick is the same as the phase at the kick
before:
φ
n
+1
=
φ
n
,
(9.9)
where
φ
n
+1
is given by (9.8). Notice that this is a transcendental equation
and has no explicit solution, but it can be solved approximately either by
numerical methods or by graphical means. We should be warned, though,
that the existence of a solution does not mean by itself that we are going to
observe it (either in an experiment or in a numerical simulation). A solution
can exist but be unstable. If we are dealing with a numerical simulation, the
slightest difference between an initial condition and the point closest to it
