Hyperchaotic attractors from a linearly controlled Lorenz system

In this paper, a four-dimensional (4D) continuous-time autonomous hyperchaotic system with only one equilibrium is introduced and analyzed. This hyperchaotic system is constructed by adding a linear controller to the second equation of the 3D Lorenz system. Some complex dynamical behaviors of the hyperchaotic system are investigated, revealing many interesting properties: (i) existence of periodic orbit with two zero Lyapunov exponents; (ii) existence of chaotic orbit with two zero Lyapunov exponents; (iii) chaos depending on initial value w0w0; (iv) chaos with only one equilibrium; and (v) hyperchaos with only one equilibrium. Finally, two complete mathematical characterizations for 4D Hopf bifurcation are derived and studied.