  The character of the time-asymptotic evolution of physical systems can have
complex, singular behavior with variation of a system parameter, particularly
when chaos is involved. A perturbation of the parameter by a small amount
$\epsilon$ can convert an attractor from chaotic to non-chaotic or vice-versa.
We call a parameter value where this can happen $\epsilon$-uncertain. The
probability that a random choice of the parameter is $\epsilon$-uncertain
commonly scales like a power law in $\epsilon$. Surprisingly, two seemingly
similar ways of defining this scaling, both of physical interest, yield
different numerical values for the scaling exponent. We show why this happens
and present a quantitative analysis of this phenomenon.
