Fractional Mathieu equation

After reviewing the concept of fractional derivative, we derive expressions for the transition curves separating regions of stability from regions of instability in the ODE:x″+(δ+ϵcost)x+cDαx=0x″+(δ+ϵcost)x+cDαx=0where DαxDαx is the order αα derivative of x(t)x(t), where 0<α<10<α<1. We use the method of harmonic balance and obtain both a lowest order approximation as well as a higher order approximation for the n=1n=1 transition curves. We also obtain an expression for the n=0n=0 transition curves.