Analysis of the Stability and Hopf Bifurcation of a Three-Dimensional System with Delays

Abstract

We propose a model of three-dimensional autonomous system with delays. We explore the dynamical behavior of the proposed autonomous system by examining bifurcation diagrams, Lyapunov exponents, equilibrium and stability, and the influence of time delay on Hopf bifurcation. A bifurcation theory is used to analyze and detail the problem. In addition, the explicit algorithm that determines the direction of Hopf bifurcation, along with the stability of bifurcating periodic, has been established. Also, there are specific operating conditions that must be met in order to achieve Hopf bifurcation. In the proposed autonomous system, we analyze the procedures for designing chaotic based systems including parameter selection, discretization of the results, as well as exploring the changing regularity of the bifurcation value. A series of numerical simulations is presented to illustrate the analytical results.