Information Technology Reference
In-Depth Information
1
0.5
0
-0.5
-1
-1
-0.5 0
Displacement
0.5
1
Figure 4.16.
The initial point lies on the intersection of a torus with the surface of section and the mapping
eventually traces out a continuous curve for the cross section of the torus.
the Poincaré surface in a smooth closed curve
and,
as more iterates are generated, the form of the curve becomes apparent and the smooth
curve is filled in. If the motion is periodic, which is to say that the orbit is closed due
to the rational nature of the frequency ratio
C
. All the iterates of
X
0
must lie on
C
X
N
. Therefore
X
0
is a
fixed point of the mapping
T
N
. If the web is integrable then such fixed points are all
that exists on the torus. No matter from where one starts, eventually the initial orbit will
close on itself. Therefore the entire curve
ω
1
/ω
2
, then
X
0
=
C
is made up of fixed points. Most curves
C
are invariant curves of the mapping because
T
maps each curve into itself,
T
(C)
=
C
.
(4.99)
The situation is quite different for non-integrable motion, since tori do not exist
for non-integrable webs. Therefore the dynamics cover the energy-conserving surface
in phase space. However, the crossing of any Poincaré surface of section will not be
confined to a torus as it was in Figure
4.16
. Subsequently we show that this lack of
confinement is a consequence of the tori having been destroyed according to KAM
theory. Trajectories in what had been regions of tori are now free to wander over larger
regions of the phase space and this is realized on
S
x
as a random-looking “splatter” of
iterates through which no smooth curve can be drawn; see, for example, Figure
4.17
.
Visual inspection of these figures does not suffice to determine whether the points are
truly distributed randomly in the plane, but, with a sufficient number of intersections,
it is clear that these points cannot be confused with the closed curves of integrable
webs. Eventually one may find small areas of the surface of section being filled up so
that a probability density function may be defined. We explore this subsequently using
numerical examples.
The surface of section is an enormously valuable tool and, when computed for a large
number of initial conditions on the same surface, it is able to give a picture of the com-
plicated phase-space structure of both integrable and non-integrable Hamiltonian webs.
But we should summarize, as did Brillouin [
10
], the weaknesses in the above discussion.
