Shilnikov chaos and Hopf bifurcation in three-dimensional differential system

In this paper, a three-dimensional differential system with only one equilibrium point is proposed. The existence of chaotic attractor in the system is verified via the homoclinic Silnikov theorem. By using the undetermined coefficient method, the homoclinic orbit of the system is determined. The Hopf bifurcation of the system is also investigated by analyzing the characteristic equation at equilibrium point and computing the first Lyapunov coefficient. Numerical simulations support the correctness of the theoretical results.