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

TL;DR

This paper analyzes the stability and bifurcation behavior of a two-term fractional differential equation with delay, identifying stability regions and how they change with parameters and delay.

Contribution

It provides a comprehensive stability and bifurcation analysis for a fractional differential equation with delay, including delay-dependent and delay-independent results.

Findings

Identified stability regions in parameter space.

Characterized stability switches and bifurcations.

Derived conditions for stability with respect to delay.

Abstract

This manuscript deals with the stability and bifurcation analysis of the equation $D^{2\alpha}x(t)+c D^{\alpha}x(t)=a x(t)+b x(t-\tau)$, 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.