arXiv subjects

Pragati Dutta

Publications and source records attributed to Pragati Dutta.

4 recordsLinked to original sources

arXiv2026

Some stability results for a Fractional Differential Equation with two delays

We investigate a nonlinear scalar Caputo fractional delay differential equation with two discrete delays and a delay-dependent feedback coefficient. Under standard Lipschitz assumptions, existence and uniqueness of solutions are established through a fixed-point argument and a fractional Gronwall inequality. The trivial equilibrium is then studied by linearization and characteristic-root analysis. When the first delay is zero, explicit delay-independent stability and instability regions are obtained, together with critical-delay conditions that account for the varying coefficient. When both delays are retained, sufficient parameter conditions for delay-independent stability and instability are derived, and an explicit sufficient threshold for the existence of a positive real characteristic root is obtained. Possible Hopf boundaries are identified through purely imaginary roots and the associated transversality condition. The influence of the fractional order on the spectral conditions is highlighted, and the analytical results are illustrated by numerical simulations and stability diagrams.

math.DS↗

arXiv2026

Analysis of Chaos and Bifurcation in Nonlinear two-delay differential equation

This paper studies how complicated and irregular behavior, known as chaos, can arise in a simple mathematical model that includes time delays. The model is a delay differential equation in which the present rate of change depends not only on the current state but also on past states at two different delay times. The system is described by \begin{equation} \dot{x}(t) = -γx(t) + g\big(x(t - τ_1)\big) - e^{-γτ_2}\, g\big(x(t - τ_1 - τ_2)\big), \end{equation} where $g(x)=k \sin{x}, \; k\in\mathbf{R}$. Here, the delays $τ_1$ and $τ_2$ represent memory effects in the system, while the sine terms introduce strong nonlinearity. Numerical simulations are used to study the system behavior for different parameter values. Chaotic motion is identified using Lyapunov exponents, Poincaré map, power sectrum analysis and phase portraits, which show irregular and unpredictable dynamics. For certain parameter ranges, the system exhibits multi-scroll chaotic attractors, in which the motion alternates among several complex patterns. The switching analysis and the parameter sensitivity analysis are provided to justify the rich dynamics. Finally, chaos is controlled by adding a simple linear feedback term, which suppresses irregular oscillations and stabilizes the system. In addition, synchronization between master and slave systems is investigated using linear state feedback control. The a delay-independent as well as a delay-dependent conditions for synchronization are derived and verified numerically. The results show that even complex delayed systems can be effectively controlled and synchronized using simple feedback techniques.

math.DS↗

arXiv2026

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

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.

math.DS↗

arXiv2024

Analysis of a Class of Two-delay Fractional Differential Equation

The differential equations involving two discrete delays are helpful in modeling two different processes in one model. We provide the stability and bifurcation analysis in the fractional order delay differential equation $D^αx(t)=a x(t)+b x(t-τ)-b x(t-2τ)$ in the $ab$-plane. Various regions of stability include stable (S), unstable (U), single stable region (SSR), and stability switch (SS). In the stable region, the system is stable for all the delay values. The region SSR has a critical value of delay that bifurcates the stable and unstable behavior. Switching of stable and unstable behaviors is observed in the SS region.

math.DS↗