Hyperchaotic attractors from a linearly controlled Lorenz system (Q1015075)

Adding a linear feedback controller to the classical Lorenz system, the authors obtain a hyperchaotic system \[ \begin{cases} \dot{x}=10\left( y-x\right) ,\\ \dot{y}=28x-y-xz+w,\\ \dot{z}=-\frac{8}{3}z+xy,\\ \dot{w}=k_{1}x+k_{2}y, \end{cases} \] where \(k_{1}\) and \(k_{2}\) are two parameters determining chaotic and hyperchaotic nature of the system. They focus attention on the study of the following properties of a new system: (a) existence of a periodic orbit with two zero Lyapunov exponents; (b) existence of a chaotic orbit with two zero Lyapunov exponents; (c) dependence of chaos on initial values of \(w_{0};\) (d) chaos with only one equilibrium; (e) hyperchaos with only one equilibrium. Furthermore, the theory of normal forms and symbolic computations are employed for the analysis of a Hopf bifurcation in the system.