Bifurcation analysis of a first time-delay chaotic system

This paper deals with the dynamic behavior of the chaotic nonlinear time delay systems of general form x˙(t)=g(x(t),x(t−τ))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\dot{x}(t)=g(x(t),x(t-\tau ))$\end{document}. We carry out stability analysis to identify the parameter zone for which the system shows a stable equilibrium response. Through the bifurcation analysis, we establish that the system shows a stable limit cycle through supercritical Hopf bifurcation beyond certain values of delay and parameters. Next, a numerical simulation of the prototype system is used to show that the system has different behaviors: stability, periodicity and chaos with the variation of delay and other parameters, which demonstrates the validity of our method. We give the single- and two-parameter bifurcation diagrams which are employed to explore the dynamics of the system over the whole parameter space.