Control of a hyperchaotic discrete system (Q5947694)

The authors study the hyperchaotic discrete system with controllable parameters, having the form \[ x_{n+1}= 1- a(x^2_n+ y^2_n)+ u_n,\quad y_{n+1}= -2a(1- 2\varepsilon) x_ny_n+ v_n. \] A local linearization of this system is considered in a small neighborhood of the desired goal \(g_n= (g^x_n, g^y_n)\). A time-varying linear feedback control law is introduced as follows \[ \begin{aligned} u_n &= g^x_{n+1}- 1+ a(g^x_n+ g^y_n)- (A_1- 2ag^x_n) (x_n- g^x_n)+ 2a g^y_n(y_n- g^y_n),\\ v_n &= g^x_{n+1}+ 2a(1- 2\varepsilon) g^x_n g^y_n+ 2a(1- 2\varepsilon) g^y_n (x_n- g^x_n)- (A_2- 2a(1- 2\varepsilon) g^x_n) (y_n- g^y_n).\end{aligned} \] In order to effectively use the above control law in the neighborhood of \(g_n\), the Lyapunov direct method is applied to estimate this neighborhood. At the end, the authors illustrate, by numerical examples, an application of the control law to solve the problem of stabilizing unstable periodic orbits and the problem of tracking an arbitrarily given periodic orbit.