Information Technology Reference
In-Depth Information
1
7
3
4
0.5
0
1
-0.5
6
2
5
-1
-1
-0.5 0 0.5
Displacement
1
Figure 4.15.
The succession of intercept points of the Poincaré surface of section. It is evident from the
numbering that the intercept points are not sequential but occur more or less at random along the
curve of the torus intersecting the plane.
If we denote the map by
T
then the sequence is generated by successive applications
of the map to the initial point as follows:
X
1
=
T
(
X
0
),
T
2
X
2
=
T
(
X
1
)
=
T
(
T
(
X
0
))
=
(
X
0
),
(4.98)
.
.
.
T
N
X
N
=
T
(
X
N
−
1
)
=
(
X
0
).
It is clear that each of the points is generated from the initial point by applying the
mapping a given number of times.
If the initial point is chosen to correspond to an orbit lying on a torus, the sequence
lies on some smooth curve resulting from the intersection of that torus with the surface
of section as indicated in Figure
4.16
. If the chosen torus is an irrational number, that
is, the frequency ratio
ω
1
/ω
2
is incommensurate, then a single orbit covers the torus
densely. In other words, the orbit is ergodic on the torus. This ergodic behavior reveals
itself on the surface of section by gradually filling in the curve with iterates until the
curve appears to be smooth. Note, however, that the ordering of the intersection points is
not sequential, but skips around the “curve” filling in at random locations. This twisting
of the intersection point is determined by the ratio of frequencies, which is chosen here
to be the square root of 13. If, on the other hand, the torus is rational, the orbit is
closed and there will be only a finite number of intersections before the iterates repeat.
Supposing that at intersection
N
we repeat the initial point
X
0
=
X
N
, the number of
iterates
N
is determined by
ω
1
/ω
2
.
The fundamental property of Hamiltonian webs is that they correspond to area-
preserving mappings. The utility of surfaces of section and the subsequent mapping
of the web dynamics on them is that the iterates reveal whether the web is integrable or
not. If the web is integrable we know that the orbit does not completely explore all of
phase space but is confined to a surface with the topology of a torus. This torus intersects
