3.7 Cantori
83
But from Eq. (3.8) we can write
A shaded =
1
1
2
∂F
∂x 1
dx 1
dt
dt +
1
1
2
∂F
∂x 0
dx 0
dt
dt = F (1, 1) − F (
1
2
,
1
2
).
(3.67)
Thus, from Eq. (3.64) the shaded area is A shaded =
K
2π 2 . The growth of A shaded
with increasing K can be seen in Fig. 3.22.
The area pumped across an M i -cycle (with winding number ω i =
N i
M i
) in one
direction is given by
W N i
M i
=
M i
j =1
[F (x
e
0(j ) , x
e
1(j ) ) − F (x
h
0(j ) , x
h
1(j ) )],
(3.68)
where x h
0(j ) (x e
0(j ) ) is the initial position of the j th hyperbolic (elliptic) fixed point of
the M i -cycle, and x h
1(j ) (x e
1(j ) ) is its position after one iteration of the map. Structures
of the type shown in Fig. 3.22, which allow phase space area to be pumped across a
line in phase space, have been called turnstiles because they behave like revolving
doors that allow a two-way flow of traffic.
Equation (3.68) is independent of the original path we took through the fixed
points. It depends only on the fact that the initial line and the line obtained after one
iteration of the map cross at the fixed points. The area, W N i
M i
, has been computed
for a number of different M i -cycles by MacKay (1982) for the universal map. Some
of his results are shown in Fig. 3.23, which shows plots of W N i
M i
as a tree in the
neighborhood of the critical noble KAM torus. The points on the tree are arranged
using the fact that [a 0 , . . . , a m+1 ] = [a 0 , . . . , a m , 1] and adding one to each side of
this equality. Note that W N i
M i
for nonnoble tori rapidly converges to a finite value,
indicating that a cantorus is present, whereas W N i
M i
for the noble KAM tori goes
to zero, indicating that no flux is present. The difference between the actions for
the stable and unstable fixed points for the rational approximates to the noble KAM
torus goes to zero.
The scaling behavior of the rational approximates implies that the flux associated
with those rational approximates also exhibits scaling behavior since the flux is
basically an area per unit time pumped across the rational approximate. We will
let K = K − K ∗ denote the distance of the parameter K from its critical value
and describe the flow of trajectories in terms of the coordinates (p, x). Then, from
Eqs. (3.50) and (3.51), we find that if we rescale K, p, and x, so that K =
K
δ ,
p =
p
β , and x =
x
α , the mapping in terms of the coordinates K , p , and x looks
exactly the same as that in terms of K, p, and x. This means that the area pumped
across a rational approximate scales in a similar manner. Thus, we can write
83
But from Eq. (3.8) we can write
A shaded =
1
1
2
∂F
∂x 1
dx 1
dt
dt +
1
1
2
∂F
∂x 0
dx 0
dt
dt = F (1, 1) − F (
1
2
,
1
2
).
(3.67)
Thus, from Eq. (3.64) the shaded area is A shaded =
K
2π 2 . The growth of A shaded
with increasing K can be seen in Fig. 3.22.
The area pumped across an M i -cycle (with winding number ω i =
N i
M i
) in one
direction is given by
W N i
M i
=
M i
j =1
[F (x
e
0(j ) , x
e
1(j ) ) − F (x
h
0(j ) , x
h
1(j ) )],
(3.68)
where x h
0(j ) (x e
0(j ) ) is the initial position of the j th hyperbolic (elliptic) fixed point of
the M i -cycle, and x h
1(j ) (x e
1(j ) ) is its position after one iteration of the map. Structures
of the type shown in Fig. 3.22, which allow phase space area to be pumped across a
line in phase space, have been called turnstiles because they behave like revolving
doors that allow a two-way flow of traffic.
Equation (3.68) is independent of the original path we took through the fixed
points. It depends only on the fact that the initial line and the line obtained after one
iteration of the map cross at the fixed points. The area, W N i
M i
, has been computed
for a number of different M i -cycles by MacKay (1982) for the universal map. Some
of his results are shown in Fig. 3.23, which shows plots of W N i
M i
as a tree in the
neighborhood of the critical noble KAM torus. The points on the tree are arranged
using the fact that [a 0 , . . . , a m+1 ] = [a 0 , . . . , a m , 1] and adding one to each side of
this equality. Note that W N i
M i
for nonnoble tori rapidly converges to a finite value,
indicating that a cantorus is present, whereas W N i
M i
for the noble KAM tori goes
to zero, indicating that no flux is present. The difference between the actions for
the stable and unstable fixed points for the rational approximates to the noble KAM
torus goes to zero.
The scaling behavior of the rational approximates implies that the flux associated
with those rational approximates also exhibits scaling behavior since the flux is
basically an area per unit time pumped across the rational approximate. We will
let K = K − K ∗ denote the distance of the parameter K from its critical value
and describe the flow of trajectories in terms of the coordinates (p, x). Then, from
Eqs. (3.50) and (3.51), we find that if we rescale K, p, and x, so that K =
K
δ ,
p =
p
β , and x =
x
α , the mapping in terms of the coordinates K , p , and x looks
exactly the same as that in terms of K, p, and x. This means that the area pumped
across a rational approximate scales in a similar manner. Thus, we can write