104
4 Chaotic Scattering
time T /2. Thus, after one period T , the phase space coordinates are mapped from
values (p 0 , x 0 ) at time t = 0 to values (p 1 , x 1 ) at time t = T , where
p 1 = p 0 + f (x 0 + p 0 T /2)
x 1 = x 0 + (p 0 + p 1 ) T /2.
(4.10)
The force exerted by the ‘kick” f (x) = 2(x 3 − x)e −x 2 is zero at the points x =
0, ±1, so the fixed points of the unperturbed system are also fixed points of the
symplectic map.
The behavior of the perturbed system is very different from that of the unperturbed system. We will want to focus on the behavior of the stable and unstable
manifolds of the symplectic map. Therefore, the initial set of points (p 0 , x 0 ) will
be a special set of points. The first step in constructing the map is to locate the
lowest order stable and unstable manifolds of the map. If we look at phase space
points near the unstable fixed points (the fixed points are not changed by the kicks),
most points escape to infinity under the action of the map. However, there is a line
of points that does not escape to infinity, but remains in the neighborhood of the
fixed points. These lines of points are called the stable and unstable manifolds of
the kicked system and can be found by a numerical search of the phase space in the
neighborhood of the fixed points.
The results of the numerical search are the stable and unstable manifolds shown
in Fig. 4.2a. The curves leaving the fixed points at (p = 0, x = −1) and (p =
0, x = +1) are the unstable manifolds. The curves approaching these fixed points
are the stable manifolds. The important point about these manifolds is that they cross
at x = 0 and thereby form a fundamental area which is the diamond shaped region
enclosed by the stable and unstable manifolds in Fig. 4.2a. The crossing points at
x = 0 are called the primary heteroclinic points of the map. We will take the curves
in Fig. 4.2a to be level 0 of the map. One iteration of the map gives the curves in
Fig. 4.2b. A second iteration of the map gives the curves in Fig. 4.2c.
Figure 4.3 gives a clearer picture of the effect of the mapping. We consider all
the points contained within the fundamental area (the shaded region) in Fig. 4.3a
(level 0). After one iteration of the map, Fig. 4.3b, the points initially inside the
fundamental area are stretched in one direction and compressed in the transverse
direction (so the area is conserved), and then folded back through the fundamental
area. If we apply the map a second time, Fig. 4.3c, this process repeats with
increasingly complex stretching and folding of the original fundamental area. With
further iterations of the map, the area of the shaded region does not change.
However, the fraction of the original shaded region that returns to the fundamental
region decreases. Note that in Fig. 4.3b, the middle section of the shaded region has
reversed its direction as it passes back through the fundamental region. Therefore,
the fixed point at (p = 0, x = 0) has become inverse hyperbolic under the action
of the map. As we continue to iterate the map for this system, the tendrils that form
from the stretching and folding of the initial stable and unstable manifolds stretch
further and further into the asymptotic region with ever greater complexity, while
maintaining their intrinsic threefold symmetry.
4 Chaotic Scattering
time T /2. Thus, after one period T , the phase space coordinates are mapped from
values (p 0 , x 0 ) at time t = 0 to values (p 1 , x 1 ) at time t = T , where
p 1 = p 0 + f (x 0 + p 0 T /2)
x 1 = x 0 + (p 0 + p 1 ) T /2.
(4.10)
The force exerted by the ‘kick” f (x) = 2(x 3 − x)e −x 2 is zero at the points x =
0, ±1, so the fixed points of the unperturbed system are also fixed points of the
symplectic map.
The behavior of the perturbed system is very different from that of the unperturbed system. We will want to focus on the behavior of the stable and unstable
manifolds of the symplectic map. Therefore, the initial set of points (p 0 , x 0 ) will
be a special set of points. The first step in constructing the map is to locate the
lowest order stable and unstable manifolds of the map. If we look at phase space
points near the unstable fixed points (the fixed points are not changed by the kicks),
most points escape to infinity under the action of the map. However, there is a line
of points that does not escape to infinity, but remains in the neighborhood of the
fixed points. These lines of points are called the stable and unstable manifolds of
the kicked system and can be found by a numerical search of the phase space in the
neighborhood of the fixed points.
The results of the numerical search are the stable and unstable manifolds shown
in Fig. 4.2a. The curves leaving the fixed points at (p = 0, x = −1) and (p =
0, x = +1) are the unstable manifolds. The curves approaching these fixed points
are the stable manifolds. The important point about these manifolds is that they cross
at x = 0 and thereby form a fundamental area which is the diamond shaped region
enclosed by the stable and unstable manifolds in Fig. 4.2a. The crossing points at
x = 0 are called the primary heteroclinic points of the map. We will take the curves
in Fig. 4.2a to be level 0 of the map. One iteration of the map gives the curves in
Fig. 4.2b. A second iteration of the map gives the curves in Fig. 4.2c.
Figure 4.3 gives a clearer picture of the effect of the mapping. We consider all
the points contained within the fundamental area (the shaded region) in Fig. 4.3a
(level 0). After one iteration of the map, Fig. 4.3b, the points initially inside the
fundamental area are stretched in one direction and compressed in the transverse
direction (so the area is conserved), and then folded back through the fundamental
area. If we apply the map a second time, Fig. 4.3c, this process repeats with
increasingly complex stretching and folding of the original fundamental area. With
further iterations of the map, the area of the shaded region does not change.
However, the fraction of the original shaded region that returns to the fundamental
region decreases. Note that in Fig. 4.3b, the middle section of the shaded region has
reversed its direction as it passes back through the fundamental region. Therefore,
the fixed point at (p = 0, x = 0) has become inverse hyperbolic under the action
of the map. As we continue to iterate the map for this system, the tendrils that form
from the stretching and folding of the initial stable and unstable manifolds stretch
further and further into the asymptotic region with ever greater complexity, while
maintaining their intrinsic threefold symmetry.