3.7 Cantori
81
Fig. 3.21 Some orbits of the
standard map for parameter
K = 1.121635. Four orbits
were started in the upper box
and stopped when they first
reached the lower box. The
black dots mark orbits that
are homoclinic to cantori with
winding numbers ω =
1
γ 2
(above) and ω =
(1+γ )
(4+3γ )
(below). These cantori act to
partially impede the flow of
trajectories in the vertical
direction (MacKay et al.
1984)
seen in Fig. 3.21, which shows diffusion in the standard map for K = 1.121635.
Although diffusion occurs across the cantorus with winding number w =
1
γ 2 , it
clearly blocks the free flow of trajectories.
The flux across a cantorus can be obtained as the limiting case of the flux across
the rational approximates to that cantorus. It is possible to determine the flux across
the rational approximates because it can be expressed in terms of an action principle
involving the elliptic and hyperbolic fixed points of the M i -cycles (Bensimon and
Kadanoff 1984; MacKay et al. 1984). Let us consider the rational approximate,
w =
0
1 , to the inverse golden mean KAM torus in the standard map, and let us draw a
line through all the periodic points of this cycle. The line through the periodic points
is (p 0 = 0, x 0 = t), where (0 ≤ t ≤ 1). After one iteration of the map, this line gets
mapped to the line (p 1 = −
K
2π sin(2πt), x 1 = t −
K
2π sin(2πt)). The shaded areas
in Fig. 3.22 include all points initially below the line p = 0 that get mapped above it
after one iteration. Since this is an area-preserving map, an equal area gets mapped
below the line p = 0. But if we are interested in the flow of particular trajectories,
they need never get mapped back below the original line. Thus the M i -cycles act as
pumps to move phase space trajectories from one part of the map to another. The
effectiveness of the M i -cycles increases as we go to larger K, as shown in Fig. 3.22
for the rational approximate w =
0
1 .
The size of the shaded areas in Fig. 3.22 can be expressed in terms of an action
principle. The Lagrangian (see Sect. 3.2.2) for the standard map can be written
81
Fig. 3.21 Some orbits of the
standard map for parameter
K = 1.121635. Four orbits
were started in the upper box
and stopped when they first
reached the lower box. The
black dots mark orbits that
are homoclinic to cantori with
winding numbers ω =
1
γ 2
(above) and ω =
(1+γ )
(4+3γ )
(below). These cantori act to
partially impede the flow of
trajectories in the vertical
direction (MacKay et al.
1984)
seen in Fig. 3.21, which shows diffusion in the standard map for K = 1.121635.
Although diffusion occurs across the cantorus with winding number w =
1
γ 2 , it
clearly blocks the free flow of trajectories.
The flux across a cantorus can be obtained as the limiting case of the flux across
the rational approximates to that cantorus. It is possible to determine the flux across
the rational approximates because it can be expressed in terms of an action principle
involving the elliptic and hyperbolic fixed points of the M i -cycles (Bensimon and
Kadanoff 1984; MacKay et al. 1984). Let us consider the rational approximate,
w =
0
1 , to the inverse golden mean KAM torus in the standard map, and let us draw a
line through all the periodic points of this cycle. The line through the periodic points
is (p 0 = 0, x 0 = t), where (0 ≤ t ≤ 1). After one iteration of the map, this line gets
mapped to the line (p 1 = −
K
2π sin(2πt), x 1 = t −
K
2π sin(2πt)). The shaded areas
in Fig. 3.22 include all points initially below the line p = 0 that get mapped above it
after one iteration. Since this is an area-preserving map, an equal area gets mapped
below the line p = 0. But if we are interested in the flow of particular trajectories,
they need never get mapped back below the original line. Thus the M i -cycles act as
pumps to move phase space trajectories from one part of the map to another. The
effectiveness of the M i -cycles increases as we go to larger K, as shown in Fig. 3.22
for the rational approximate w =
0
1 .
The size of the shaded areas in Fig. 3.22 can be expressed in terms of an action
principle. The Lagrangian (see Sect. 3.2.2) for the standard map can be written