Hopf bifurcation and its control in a 3D autonomous system

The first section of this paper discusses the stability and Hopf bifurcation for a new dynamical system using stability theory, the center manifold as well as normal form theory. To verify the analytical results, numerical simulations are performed. The second section focuses on controlling the Hopf bifurcation with a robust controller capable of handling a wide range of parameter values. By fine tuning the control parameters, the controller ensures that Hopf bifurcation occurred at P0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{0}$$\end{document}. Furthermore, we postpone the Hopf bifurcation at P+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{+}$$\end{document} by adjusting the control parameters.