  Assuming a Friedmann universe which evolves with a power-law scale factor,
$a=t^{n}$, we analyse the phase space of the system of equations that describes
a time-varying fine structure 'constant', $\alpha$, in the
Bekenstein-Sandvik-Barrow-Magueijo generalisation of general relativity. We
have classified all the possible behaviours of $\alpha (t)$ in ever-expanding
universes with different $n$ and find new exact solutions for $\alpha (t)$. We
find the attractors points in the phase space for all $n$. In general, $\alpha
$ will be a non-decreasing function of time that increases logarithmically in
time during a period when the expansion is dust dominated ($n=2/3$), but
becomes constant when $n>2/3$. This includes the case of negative-curvature
domination ($n=1$). $\alpha $ also tends rapidly to a constant when the
expansion scale factor increases exponentially. A general set of conditions is
established for $\alpha $ to become asymptotically constant at late times in an
expanding universe.
