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Figure 4.12.
A trajectory in a three-dimensional phase space intersects a plane at a number of points that are
labeled by the plane's coordinates. These points contain the measured information about the
trajectory. Here an invariant torus is defined by the actions
J
1
and
J
2
and angles
θ
1
(
t
)
=
1
t
+
θ
1
(
)
θ
2
(
t
)
=
2
t
+
θ
2
(
).
0
and
0
The trajectory is periodic if the ratio of the
frequencies
1
/
2
is rational and the orbit intersects the plane at a finite number of points. The
trajectory is quasi-periodic if the ratio of the frequencies is irrational and the orbit intersects the
plane at a continuum of points that eventually sculpts out the torus.
If the quantities
X
n
and
Y
n
can be uniquely determined in terms of their values in the
next iteration,
X
n
+
1
and
Y
n
+
1
, meaning a functional relation of the form
X
n
=
G
1
(
X
n
+
1
,
Y
n
+
1
),
(4.82)
Y
n
=
G
2
(
X
n
+
1
,
Y
n
+
1
),
then the two-dimensional map is invertible. If
n
is the time index of the map then invert-
ibility is equivalent to time reversibility, so these maps are reversible in time, whereas
the one-humped maps are not. The maps discussed here are analogous to Newtonian
dynamics.
An example of a simple two-dimensional map is given by
X
n
+
1
=
f
(
X
n
)
+
Y
n
,
(4.83)
Y
n
+
1
=
β
X
n
,
where, if the mapping function
f
0, the dynamic equation
reduces to the one-dimensional map of the last section. For any non-zero
(
·
)
is not invertible and
β
=
β
, however,
the map (
4.83
) is invertible since the inverse function can be defined as
X
n
=
Y
n
+
1
/β,
(4.84)
Y
n
=
X
n
+
1
−
f
(
Y
n
+
1
/β).
In this way a non-invertible map can be transformed into an invertible one, or, said
differently, extending the phase space from one to two dimensions lifts the ambiguity
present in the one-dimensional map.
Another property of interest is that the volume of phase space occupied by a
dynamical web remains unchanged (conserved) during evolution. For example, the
cream in your morning coffee can be stirred until it spreads from a concentrated blob
of white in a black background into a uniformly tan mixture. The phase-space volume
occupied by the cream does not change during this process of mixing. In the same way
a volume
V
n
remains unchanged during iterations if the Jacobian of the iteration
