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

Fractional Order Periodic Maps: Stability Analysis and Application to the Periodic-2 Limit Cycles in the Nonlinear Systems