Stability and Bifurcation Analysis of Fractional Delay Differential Equation with a Delay-dependent Coefficient
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arXiv
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| Format: | Preprint |
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2026
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| author | Dutta, Pragati Bhalekar, Sachin |
| author_facet | Dutta, Pragati Bhalekar, Sachin |
| contents | This paper investigates the stability of different regions in the $(k,γ)$-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 $k = g'(0)$. 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 $τ_1 = 0$ and $τ_2 \ge 0$. Furthermore, a general result is established for the case $τ_1 > 0$, $τ_2 \ge 0$, which holds for all values of $α$ and $τ_1$.
In addition, illustrative examples are provided in the form of stability diagrams in the $(τ_1,τ_2)$-plane for fixed values of $α$, $k$, and $γ$. These diagrams are generated using appropriate numerical methods to visualize the stability regions and to support the theoretical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_04822 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stability and Bifurcation Analysis of Fractional Delay Differential Equation with a Delay-dependent Coefficient Dutta, Pragati Bhalekar, Sachin Dynamical Systems 34K20, 34K18, 37C75, 37C75, 34K18 This paper investigates the stability of different regions in the $(k,γ)$-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 $k = g'(0)$. 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 $τ_1 = 0$ and $τ_2 \ge 0$. Furthermore, a general result is established for the case $τ_1 > 0$, $τ_2 \ge 0$, which holds for all values of $α$ and $τ_1$. In addition, illustrative examples are provided in the form of stability diagrams in the $(τ_1,τ_2)$-plane for fixed values of $α$, $k$, and $γ$. These diagrams are generated using appropriate numerical methods to visualize the stability regions and to support the theoretical results. |
| title | Stability and Bifurcation Analysis of Fractional Delay Differential Equation with a Delay-dependent Coefficient |
| topic | Dynamical Systems 34K20, 34K18, 37C75, 37C75, 34K18 |
| url | https://arxiv.org/abs/2605.04822 |