Stability and Hopf bifurcation of a Lorenz-like system

Hopf bifurcation is one of the important dynamical behaviors. It could often cause some phenomena, such as quasiperiodicity and intermittency. Consequently, chaos will happen due to such dynamical behaviors. Since chaos appears in the Lorenz-like system, to understand the dynamics of such system, Hopf bifurcation will be explored in this paper. First, the stability of equilibrium points is presented. Then Hopf bifurcation of the Lorenz-like system is investigated. By applying the normal form theory, the conditions guaranteeing the Hopf bifurcation are derived. Further, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are also presented. Finally, numerical simulations are given to verify the theoretical analysis. It is found that Hopf bifurcation could happen when conditions are satisfied. The stable bifurcating periodic orbit is displayed. Chaos will also happen when parameter further increases.