Consider the configuration shown in Fig. 2. We wish to
determine which exit a particle takes to leave the system when it is
incident from the left at y = b and its initial direction of approach
makes an angle 
with respect to the y-axis. These are the two
input variables which we will be varying to see how they affect the
outcome of the trajectories. We define the scattering region as the area bounded by the
equilateral triangle and the disks (it is shown shaded in
Fig. 2).
We are only interested in trajectories whose paths eventually enter the
scattering region. Once a trajectory enters the scattering region, it will
leave the region through one of the exits labeled in Fig. 2.
(For b and
randomly chosen, there is zero probability for the
trajectory to stay inside the scattering region for all time.)
To visually differentiate which initial condition leads to which exit, we have
color-coded the phase space of initial conditions as follows: if a
trajectory with initial condition defined by
(b,) leaves through
exit I (Fig. 2), then the point (b,) will be colored red.
Similarly, initial conditions whose trajectories leave through exit II
will be colored green, and ones that leave through exit III will be colored blue.
Initial conditions whose trajectories never enter the scattering region will
be colored white.
In Fig. 3 , we see pictures of the basins of attraction for the different exits at various magnification levels. From the pictures, it seems that no matter how much we amplify the magnification, there does not seem to be any regions where a boundary separates only two basins. It appears that there is always a cluster of three colors near the boundaries. This gives a clear (although nonrigorous) indication that the basin boundaries are Wada.
Before we proceed to show more rigorously that the basin boundaries for this
system possess the Wada property (see Sec. 4), we shall compute
the uncertainty dimension of the basin boundaries to see how tiny perturbations
in the initial conditions affect the predictability of the final state.
Following a method proposed by Grebogi et al. [1983],
we randomly select a large number N of initial conditions
(b,)
from some specified region in phase space and determine their exit modes.
The exit mode calculations are now repeated for initial conditions under
-perturbation of b, namely
(b+,) and (b-,). If the trio of initial
conditions, (b-,),(b,) and (b+,) results in trajectories that leave through the same exit, then the original initial
condition (b,) is considered ``certain'' under -perturbation
of b; otherwise, it is deemed ``uncertain''. For a given perturbation
, we can compute the fraction
f() of initial conditions
which leads to uncertain final states. In general, f() scales with as
f() 
, where
is called the uncertainty
exponent. The uncertainty dimension of the basin boundaries is then defined as
d = D - where
D is the dimension of phase space [Grebogi et al., 1983;
MacDonald et al., 1985]. In this case, D=2.
Figs. 4(b)-(d) show the log-log plots of
f() vs for three different regions in phase space as outlined in Fig. 4(a). It is seen that, for all three of these regions, the plots are well
approximated by a power law, and we obtained a mean value of
0.27 for the
uncertainty exponent, or d 1.73
for the uncertainty dimension. The implication of such a low value for the
uncertainty exponent is quite striking. For example, increasing the precision
in the specification of initial conditions by a factor of 10 reduces
f() by only about 50%
(10-0.27 0.54).
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