4.3 Scattering Chaos in a Magnetic Dipole
113
bowl, they need not come back through the PSS in that region. They can rapidly
escape to the asymptotic region.
We expect interesting behavior to occur when periodic orbits form in the PSS.
The orbits initially in the neighborhood of the potential bowl remain in that region
of the phase space. For energies E > E U = 0.081 . . . , no periodic orbits exist in
the PSS in the neighborhood of the potential energy bowl. As the energy is lowered
to the value E = E U = 0.081 . . . , a bifurcation occurs, creating a pair of period 1
periodic orbits in the PSS, one elliptic () and the other hyperbolic (γ ) (Jung and
Scholz 1988; Rückerl 1994). These points lie on the symmetry line ˙
ρ = 0. As energy
is lowered further, the elliptic fixed point, , changes to inverse hyperbolic and the
phase space in the neighborhood of the potential energy bowl becomes more chaotic.
The fixed point γ is associated with the saddle, and its position approaches the point
ρ = 2, z = 0 as the energy E→E S . It corresponds to an unstable periodic orbit that
oscillates back and forth along the potential energy ridge that lies to the right of
the potential energy bowl. A particle incident from the asymptotic region can only
experience scattering chaos if its energy lies in the interval 0.03125 < E < 0.081.
In Fig. 4.10a, we show a PSS for energy E = 0.079 in the region of phase
space containing the pair of period 1 periodic orbits mentioned above. The position
of the elliptic periodic orbit, P e , is surrounded by KAM tori. The location of the
hyperbolic period 1 periodic orbit, P u , lies at the crossing point of its stable and
unstable manifolds, W s and W u , respectively. Only a small segment of the unstable
manifold is shown, but some of the homoclinic tendrils of the stable manifold are
shown. If the same number of tendrils of the unstable manifold had been drawn,
the figure would have reflection symmetry about the ˙
ρ = 0 axis. The homoclinic
tendrils of the stable manifold are obtained by taking a small segment of the stable
manifold near the fixed point and iterating the PSS backward in time. Some tendrils
extend into the asymptotic region.
Also shown in Fig. 4.10a is a dotted line that runs parallel to the unstable
manifold and crosses some of the tendrils of the stable manifold. This dotted line
is obtained by starting at the initial time, t = 0, with a line of initial conditions in
the asymptotic region and following it in time until it crosses the surface of section
in the neighborhood of the potential bowl. The initial conditions used to obtain the
dotted line have momentum parallel to the ρ − φ plane and directed toward the
origin. All initial points are the same radial distance, ρ = ρ 0 , from the z-axis in the
asymptotic region but have a range of values of z. Notice that, when they intersect
the PSS, some of these points lie inside the region formed by the tendrils. These
points can be trapped for a long time inside the reaction region as they make their
way through the complex network of homoclinic tangles. If they happen to lie on the
stable manifold itself, they will never escape. This can happen if a tendril crosses
the line of initial points in the asymptotic region.
In Fig. 4.10b, we show the delay time of scattered trajectories with E = 0.079.
All the trajectories initially have momenta parallel to the ρ − φ plane and directed
toward the origin. All initial conditions are in the asymptotic region with the
same value of ρ = ρ 0 and a range of initial values z = z i in the interval
1.3577 < z i < 1.3583. The delay time is defined as T = T a − T h , where T a is
113
bowl, they need not come back through the PSS in that region. They can rapidly
escape to the asymptotic region.
We expect interesting behavior to occur when periodic orbits form in the PSS.
The orbits initially in the neighborhood of the potential bowl remain in that region
of the phase space. For energies E > E U = 0.081 . . . , no periodic orbits exist in
the PSS in the neighborhood of the potential energy bowl. As the energy is lowered
to the value E = E U = 0.081 . . . , a bifurcation occurs, creating a pair of period 1
periodic orbits in the PSS, one elliptic () and the other hyperbolic (γ ) (Jung and
Scholz 1988; Rückerl 1994). These points lie on the symmetry line ˙
ρ = 0. As energy
is lowered further, the elliptic fixed point, , changes to inverse hyperbolic and the
phase space in the neighborhood of the potential energy bowl becomes more chaotic.
The fixed point γ is associated with the saddle, and its position approaches the point
ρ = 2, z = 0 as the energy E→E S . It corresponds to an unstable periodic orbit that
oscillates back and forth along the potential energy ridge that lies to the right of
the potential energy bowl. A particle incident from the asymptotic region can only
experience scattering chaos if its energy lies in the interval 0.03125 < E < 0.081.
In Fig. 4.10a, we show a PSS for energy E = 0.079 in the region of phase
space containing the pair of period 1 periodic orbits mentioned above. The position
of the elliptic periodic orbit, P e , is surrounded by KAM tori. The location of the
hyperbolic period 1 periodic orbit, P u , lies at the crossing point of its stable and
unstable manifolds, W s and W u , respectively. Only a small segment of the unstable
manifold is shown, but some of the homoclinic tendrils of the stable manifold are
shown. If the same number of tendrils of the unstable manifold had been drawn,
the figure would have reflection symmetry about the ˙
ρ = 0 axis. The homoclinic
tendrils of the stable manifold are obtained by taking a small segment of the stable
manifold near the fixed point and iterating the PSS backward in time. Some tendrils
extend into the asymptotic region.
Also shown in Fig. 4.10a is a dotted line that runs parallel to the unstable
manifold and crosses some of the tendrils of the stable manifold. This dotted line
is obtained by starting at the initial time, t = 0, with a line of initial conditions in
the asymptotic region and following it in time until it crosses the surface of section
in the neighborhood of the potential bowl. The initial conditions used to obtain the
dotted line have momentum parallel to the ρ − φ plane and directed toward the
origin. All initial points are the same radial distance, ρ = ρ 0 , from the z-axis in the
asymptotic region but have a range of values of z. Notice that, when they intersect
the PSS, some of these points lie inside the region formed by the tendrils. These
points can be trapped for a long time inside the reaction region as they make their
way through the complex network of homoclinic tangles. If they happen to lie on the
stable manifold itself, they will never escape. This can happen if a tendril crosses
the line of initial points in the asymptotic region.
In Fig. 4.10b, we show the delay time of scattered trajectories with E = 0.079.
All the trajectories initially have momenta parallel to the ρ − φ plane and directed
toward the origin. All initial conditions are in the asymptotic region with the
same value of ρ = ρ 0 and a range of initial values z = z i in the interval
1.3577 < z i < 1.3583. The delay time is defined as T = T a − T h , where T a is