Stability and Bifurcation Analysis of Fractional Delay Differential Equation with a Delay-dependent Coefficient

Abstract

This paper investigates the stability of different regions in the $(k,\gamma )$-plane for a class of fractional delay differential equations given by \begin{equation} D^α x(t) = -γx(t) + g\big(x(t - τ_1)\big) - e^{-γτ_2}\, g\big(x(t - τ_1 - τ_2)\big), \qquad 0 < α\le 1, \end{equation} where $k=g^{\prime }(0)$. The primary focus is on the stability of the trivial equilibrium of the corresponding linearized system. A detailed stability and bifurcation analysis is carried out for the particular case ${\tau }_1=0$ and ${\tau }_2\ge 0$. Furthermore, a general result is established for the case ${\tau }_1>0$, ${\tau }_2\ge 0$, which holds for all values of $\alpha$ and ${\tau }_1$. In addition, illustrative examples are provided in the form of stability diagrams in the $({\tau }_1,{\tau }_2)$-plane for fixed values of $\alpha$, $k$, and $\gamma$. These diagrams are generated using appropriate numerical methods to visualize the stability regions and to support the theoretical results.

23 pages, 50 figures