  Results are presented from numerical simulations of the flat-space nonlinear
Klein-Gordon equa- tion with an asymmetric double-well potential in spherical
symmetry. Exit criteria are defined for the simulations that are used to help
understand the boundaries of the basins of attraction for Gaussian "bubble"
initial data. The first exit criteria, based on the immediate collapse or
expan- sion of bubble radius, is used to observe the departure of the scalar
field from a static intermediate attractor solution. The boundary separating
these two behaviors in parameter space is smooth and demonstrates a
time-scaling law with an exponent that depends on the asymmetry of the
potential. The second exit criteria differentiates between the creation of an
expanding true-vacuum bubble and dispersion of the field leaving the false
vacuum; the boundary separating these basins of attraction is shown to
demonstrate fractal behavior. The basins are defined by the number of bounces
that the field undergoes before inducing a phase transition. A third, hybrid
exit criteria is used to determine the location of the boundary to arbitrary
precision and to characterize the threshold behavior. The possible effects this
behavior might have on cosmological phase transitions are briefly discussed.
