4.4 Model of Chlorine Ion in a Radiation Field
119
Fig. 4.14 (a) Four iterations
of a line of initial points
(initially in interval
−10.5≤x≤10.1) just before
the line of points enters the
fundamental region R.
(b) Delay time of the points
before they leave (transmitted
or reflected) the fundamental
region. (c) Number of steps
taken inside the fundamental
region (only points with less
than 20 steps shown)
(reproduced from Lin et al.
2011)
4.4.3 Symbolic Dynamics
A symbolic dynamics for this scattering process can be constructed by examining
the structure of the gaps that are cut into the unstable manifolds, n–q and p–m, by the
tendrils formed by the stable manifolds, r–p and s–n, as they are iterated backward
in time. In Fig. 4.15a, we show the first iteration of the stable manifolds backward
in time. The shaded regions contain points from the asymptotic regions that enter
and cut through the fundamental region R on this first iteration of the PSS. The
shaded regions cut two segments out of the unstable manifold of F L and cut one
segment out of the unstable manifold of F R . We label the three remaining pieces
of the unstable manifold of F L as “A”, “B”, and “C”. We label the two remaining
pieces of the unstable manifold of F R as “+” and “C”. In Fig. 4.15b, we show the
result of two iterations of the PSS. In this second iteration, we need to introduce
two additional symbols to fully determine the symbolic dynamics. The left-most
symbols of the unstable manifold of F L are again “A”, “B”, and “C” because these
segments are formed in exactly that same way as in Fig. 4.15a.
119
Fig. 4.14 (a) Four iterations
of a line of initial points
(initially in interval
−10.5≤x≤10.1) just before
the line of points enters the
fundamental region R.
(b) Delay time of the points
before they leave (transmitted
or reflected) the fundamental
region. (c) Number of steps
taken inside the fundamental
region (only points with less
than 20 steps shown)
(reproduced from Lin et al.
2011)
4.4.3 Symbolic Dynamics
A symbolic dynamics for this scattering process can be constructed by examining
the structure of the gaps that are cut into the unstable manifolds, n–q and p–m, by the
tendrils formed by the stable manifolds, r–p and s–n, as they are iterated backward
in time. In Fig. 4.15a, we show the first iteration of the stable manifolds backward
in time. The shaded regions contain points from the asymptotic regions that enter
and cut through the fundamental region R on this first iteration of the PSS. The
shaded regions cut two segments out of the unstable manifold of F L and cut one
segment out of the unstable manifold of F R . We label the three remaining pieces
of the unstable manifold of F L as “A”, “B”, and “C”. We label the two remaining
pieces of the unstable manifold of F R as “+” and “C”. In Fig. 4.15b, we show the
result of two iterations of the PSS. In this second iteration, we need to introduce
two additional symbols to fully determine the symbolic dynamics. The left-most
symbols of the unstable manifold of F L are again “A”, “B”, and “C” because these
segments are formed in exactly that same way as in Fig. 4.15a.