Stability and Bifurcation Analysis for the Dynamical Model of a New Three-dimensional Chaotic System

Chaotic systems have important applications in secure communications. Therefore, it is of great theoretical and practical significance to study the dynamics of this class of system. This paper is devoted to investigating the dynamic behaviors of a three dimensional chaotic system. With both analytical and numerical methods, the nonlinear dynamic characteristics including stability and bifurcations of this new three dimensional chaotic system are investigated in this paper. It is presented that this system has symmetry and invariance. All the equilibriums of the system and their stability are studied in detail for different values of system parameters. It is presented that there may exist three equilibriums for this system. The stability conditions of these equilibriums are obtained with the Routh–Hurwitz criterion. Using the Hopf bifurcation theorem and the first Lyapunov coefficient, the system condition and type of Hopf bifurcation for this system is obtained analytically. It is demonstrated that there may exist subcritical Hopf bifurcations under certain system parameters for this system. Using the Runge-Kutta method, numerical simulations including phase portraits and time history curves are also given, which verify the analytical results. The results obtained here can provide some guidance for the analysis and design of secure communication systems.