Hopf bifurcation of forced Chen system and its stability via adaptive control with arbitrary parameters

In this article, forced Chen system is analyzed for nonlinear dynamical behavior. The chaotic behavior of forced Chen system is verified by phase portraits and sensitivity dependence of system upon initial condition. Hopf bifurcation for the complex system is derived and theorem of first Lyapunov coefficient is used to investigate the type of Hopf bifurcation. It is further shown that Hopf bifurcation exists only on two equilibrium points for the proposed chaotic model. In addition, an adaptive control technique is used to control unpredictable behavior for the forced Chen system. Global stability is achieved by constructing an energy type function through Lyapunov theory, whereas its error dynamics is used to synchronize two identical forced Chen systems. Numerical simulation results are used to validate analytical results given in this article and also to demonstrate effectiveness of the considered chaotic system.