112
4 Chaotic Scattering
Fig. 4.9 Contour plot of the
potential energy V (ρ, z) as a
function of ρ and z. Some
specific contour lines of
constant potential energy with
values of V (ρ, z) are shown
Jung and Scholz (1988)
Fig. 4.10 (a) A Poincaré
surface of section, ˙
ρ versus ρ,
for surface z = 0, ˙
z > 0 and
energy E = 0.079. Some
homoclinic tendrils for the
stable manifold, W s , and a
short segment of the unstable
manifold, W u , are shown. (b)
The time delay of a sequence
of trajectories (the dotted line
in (a)) with energy
E = 0.079, ρ = ρ 0 , and a
range of initial values z = z i
(Jung and Scholz 1988)
region of influence of the dipole potential) and eventually returns to the asymptotic
region in an altered state. For energies E < E L =
1
32 = 0.03125, a particle cannot
enter the potential energy bowl from the asymptotic region.
We can study the behavior of orbits inside the potential energy bowl by means
of a Poincaré surface of section (PSS). In Fig. 4.10a, we plot the phase space
coordinates ( ˙
ρ, ρ) each time a trajectory crosses the z = 0 surface with ˙
z > 0. If
we start with a line of initial conditions in the neighborhood of the potential energy