  We address whether Pad\'e-summations of the $\bar{MS}$ QCD $\beta$-function
for a given number of flavours exhibit an infrared-stable fixed point, or
alternatively, an infrared attractor of a double valued couplant as noted by
Kogan and Shifman for the case of supersymmetric gluodynamics. Below an
approximant-dependent flavour threshold $(6 \leq n_f \leq 8)$, we find that
Pad\'e-summation $\beta$-functions incorporating $[2|1], [1|2], [2|2], [1|3]$,
and $[3|1]$ approximants always exhibit a positive pole prior to the occurrence
of their first positive zero, precluding any identification of this first
positive zero as an infrared-stable fixed point of the $\beta$- function. This
result is shown to be true regardless of the magnitude of the presently-unknown
five-loop $\beta$-function contribution. Moreover, the pole in question
suggests the occurrence of dynamics in which both a strong and an
asymptotically-free phase share a common infrared attractor. We briefly discuss
the possible relevance of infrared-attractor dynamics to the success of recent
calculations of the glueball mass spectra in QCD with $N_c \to \infty$ via
supergravity. As $n_f$ increases above an approximant-dependent flavour
threshold, Pad\'e-summation $\beta$-functions incorporating $[2|2], [1|3]$, and
$[3|1]$ approximants exhibit dynamics controlled by an infrared-stable fixed
point over a widening domain of the five-loop $\bar{MS}$ $\beta$-function
parameter $(\beta_4/\beta_0)$. Above this threshold, all approximants
considered exhibit infrared-stable fixed points that decrease in magnitude with
increasing flavour number.
