  We consider the stability of periodic map with period-$2$ in linear
fractional difference equations where the function is $f(x)=ax$ at even times
and $f(x)=bx$ at odd times. The stability of such a map for an integer order
map depends on product $ab$. The conditions are much complex for fractional
maps and depend on $ab$ as well as $a+b$. There are no superstable period-2
orbits. These conditions are useful in obtaining stability conditions of
asymptotically periodic orbits with period-$2$ in the nonlinear case. The
stability conditions are demonstrated numerically. The formalism can be
generalized to higher periods.
