4.4 Model of Chlorine Ion in a Radiation Field
117
Fig. 4.12 A PSS with p and
x plotted each time φ =
π
2 .
(a) The curves n − u and
s − n ( r − p and p − v) are
segments of the unstable and
stable manifolds of F L (F R ).
The fundamental region is
enclosed by the curves n − u,
s − n, r − p and p − v. (b)
Segments of the stable
manifolds r–p (solid line) and
n–s (dashed line) are
integrated backward in time
for four iterations of the PSS
(reproduced from Lin et al.
2011)
We can distinguish points associated with the asymptotic region from those
associated with the fundamental region. This becomes clear in Fig. 4.13a, b,
where the shaded regions with positive indices contain points that lie outside the
fundamental region. Iterations of these shaded regions are shown for times ranging
from t = −3 to t = +3 (seven iterations of the PSS). As the map is iterated
backward in time, the shaded regions form more and more complex filaments that
cut across the fundamental region. As the shaded regions are iterated forward in
time, they approach their respective asymptotic regions. In Fig. 4.13a, we follow the
point α n (−3≤n≤3) which is incident from the right at time n = −3, but happens
initially to be inside one of the tendrils in the asymptotic region at that time. In
Fig. 4.13b, we follow the point β n (−3≤n≤4), which is incident from the left at
time n = −3 and is initially inside a tendril in the asymptotic region at that time.
Careful inspection of Figs. 4.12b and 4.13a, b shows that the “backward in
time” filaments from the two stable manifolds become interlaced in the tendrils that
extend backward in time to the upper left, and backward in time to the lower right.
A particle, incident from the left, that initially lies in one of these “backward in time”
tendrils can either be transmitted through the scattering region toward x = +∞ or
it can be reflected from the scattering region toward x = −∞, depending whether it
initially lies in a filament of the type shown in Fig. 4.13a or that shown in Fig. 4.13b.
117
Fig. 4.12 A PSS with p and
x plotted each time φ =
π
2 .
(a) The curves n − u and
s − n ( r − p and p − v) are
segments of the unstable and
stable manifolds of F L (F R ).
The fundamental region is
enclosed by the curves n − u,
s − n, r − p and p − v. (b)
Segments of the stable
manifolds r–p (solid line) and
n–s (dashed line) are
integrated backward in time
for four iterations of the PSS
(reproduced from Lin et al.
2011)
We can distinguish points associated with the asymptotic region from those
associated with the fundamental region. This becomes clear in Fig. 4.13a, b,
where the shaded regions with positive indices contain points that lie outside the
fundamental region. Iterations of these shaded regions are shown for times ranging
from t = −3 to t = +3 (seven iterations of the PSS). As the map is iterated
backward in time, the shaded regions form more and more complex filaments that
cut across the fundamental region. As the shaded regions are iterated forward in
time, they approach their respective asymptotic regions. In Fig. 4.13a, we follow the
point α n (−3≤n≤3) which is incident from the right at time n = −3, but happens
initially to be inside one of the tendrils in the asymptotic region at that time. In
Fig. 4.13b, we follow the point β n (−3≤n≤4), which is incident from the left at
time n = −3 and is initially inside a tendril in the asymptotic region at that time.
Careful inspection of Figs. 4.12b and 4.13a, b shows that the “backward in
time” filaments from the two stable manifolds become interlaced in the tendrils that
extend backward in time to the upper left, and backward in time to the lower right.
A particle, incident from the left, that initially lies in one of these “backward in time”
tendrils can either be transmitted through the scattering region toward x = +∞ or
it can be reflected from the scattering region toward x = −∞, depending whether it
initially lies in a filament of the type shown in Fig. 4.13a or that shown in Fig. 4.13b.