  We analyze situations where a saddle-node bifurcation occurs on a fractal
basin boundary. Specifically, we are interested in what happens when a system
parameter is slowly swept in time through the bifurcation. Such situations are
known to be indeterminate in the sense that it is difficult to predict the
eventual fate of an orbit that tracks the pre-bifurcation node attractor as the
system parameter is swept through the bifurcation. In this paper we investigate
the scaling of (1) the fractal basin boundary of the static (i.e., unswept)
system near the saddle-node bifurcation, (2) the dependence of the orbit's
final destination on the sweeping rate, (3) the dependence of the time it takes
for an attractor to capture a swept orbit on the sweeping rate, and (4) the
dependence of the final attractor capture probability on the noise level. With
respect to noise, our main result is that the effect of noise scales with the
5/6 power of the parameter drift rate. Our approach is to first investigate all
these issues using one-dimensional map models. The simplification of treatment
inherent in one dimension greatly facilitates analysis and numerical
experiment, aiding us in obtaining the new results listed above. Following our
one-dimensional investigations, we explain that these results can be applied to
two-dimensional systems. We show, through numerical experiments on a
periodically forced second order differential equation example, that the
scalings we have found also apply to systems that result in two dimensional
maps.
