  We solve the equations of two-dimensional hydrodynamics describing a
circumbinary disk accreting onto an eccentric, equal-mass binary. We compute
the time rate of change of the binary semi-major axis $a$ and eccentricity $e$
over a continuous range of eccentricities spanning $e=0$ to $e=0.9$. We find
that binaries with initial eccentricities $e_0\lesssim 0.1$ tend to $e=0$,
where the binary semi-major axis expands. All others are attracted to $e
\approx 0.4$, where the binary semi-major axis decays. The $e \approx 0.4$
attractor is caused by a rapid change in the disk response from an apsidally
locked, nearly symmetric state to a precessing asymmetric state. The state
change causes the time rates of change $\dot{a}$ and $\dot{e}$ to steeply
change sign at the same critical eccentricity resulting in an attracting
solution where $\dot{a} = \dot{e} = 0$. This does not, however, result in a
stalled, eccentric binary. Due to a finite transition time between disk states,
the binary eccentricity continually overshoots the attracting eccentricity
resulting in a constant drift of the semi-major axis. For fiducial disk
parameters, binaries with $e_0 \gtrsim 0.1$ evolve towards and then oscillate
around $e \approx 0.4$ where they steadily shrink in semi-major axis. Because
unequal mass binaries grow towards equal mass through preferential accretion,
our results are applicable to a wide range of initial binary mass ratios.
Hence, these findings merit further investigations of this disk transition;
understanding its dependence on disk parameters is vital for determining the
fate of binaries undergoing orbital evolution with a circumbinary disk.
