Stabilizing unstable fixed points of discrete chaotic systems via quasi-sliding mode method

The problem of stabilizing unstable fixed points of nonlinear discrete chaotic systems, subjected to bounded model uncertainties, is investigated in this article. The theory is then generalized to include any dth-order fixed point of the system. To design a suitable controller, the theory of quasi-sliding mode control is modified and applied. Sufficient conditions for the convergence of the control algorithm are theoretically derived and it is shown that the error trajectories converge toward a bounded region around zero where the measure of the steady-state error band depends on the magnitude of the system uncertainties. As a case study, the proposed method is applied to the Henon map to stabilize its first, second, and fourth-order unstable fixed points. Simulation results show the high performance of the control technique in quenching the chaos in the presence of uncertainties.