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) = -\gamma x(t) + g\big(x(t - \tau_1)\big) - e^{-\gamma \tau_2}, g\big(x(t - \tau_1 - \tau_2)\big), \qquad 0 < \alpha \le 1, \end{equation} where $g(x)=k \sin{x}, k\in\mathbf{R}$. Here, the delays $\tau_1$ and $\tau_2$ 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 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. 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, and a delay-independent sufficient condition for synchronization is derived and verified numerically. The results show that even complex delayed systems can be effectively controlled and synchronized using simple feedback techniques. The study is further extended to a fractional-order version of the system to examine the influence of memory effects, where it is observed that chaotic behavior can persist even for lower fractional orders.