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

Pragati Dutta; Sachin Bhalekar

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arXiv · 2604.07924

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

Pragati Dutta·Sachin Bhalekar

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Abstract

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), \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, Poincar\'e 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.

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BibTeX RIS

Pragati Dutta, Sachin Bhalekar. 2026-04-09. Analysis of Chaos and Bifurcation in Nonlinear two-delay differential equation. https://arxiv.org/abs/2604.07924

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