Stability and Bifurcation Analysis of Two-Term Fractional Differential Equation with Delay

This manuscript deals with the stability and bifurcation analysis of the equation D2αx(t)+c Dαx(t)=a x(t)+b x(t-τ), where 0<α<1 and τ>0. We sketch the boundaries of various stability regions in the parameter plane under different conditions on α and b. First, we provide the stability analysis of this equation with τ=0. Change in the stability of the delayed counterpart is possible only when the characteristic roots cross the imaginary axis. This leads to various delay-independent as well as delay-dependent stability results. The stability regions are bifurcated on the basis of the following behaviors with respect to the delay τ viz. stable region for all τ>0, unstable region, single stable region, stability switch, and instability switch.