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

Abstract

This manuscript deals with the stability and bifurcation analysis of the equation $D^{2\alpha }x(t)+c{D}^{\alpha }x(t)=ax(t)+bx(t-\tau )$, where $0<\alpha <1$ and $\tau >0$. We sketch the boundaries of various stability regions in the parameter plane under different conditions on $\alpha$ and $b$. First, we provide the stability analysis of this equation with $\tau =0$. 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 $\tau$ viz. stable region for all $\tau >0$, unstable region, single stable region, stability switch, and instability switch.

55 pages, 100 figures