Control of stochastic chaos using sliding mode method

Stabilizing unstable periodic orbits of a deterministic chaotic system which is perturbed by a stochastic process is studied in this paper. The stochastic chaos is modeled by exciting a deterministic chaotic system with a white noise obtained from derivative of a Wiener process which eventually generates an Ito differential equation. It is also assumed that the chaotic system being studied has some model uncertainties which are not random. The sliding mode controller with some modifications is used for stochastic chaos suppression. It is shown that the system states converge to the desired orbit in such a way that the error covariance converges to an arbitrarily small bound around zero. As some case studies, the stabilization of 1-cycle and 2-cycle orbits of chaotic Duffing and Φ6Φ6 Van der Pol systems is investigated by applying the proposed method to their corresponding stochastically perturbed systems. Simulation results show the effectiveness of the method and the accuracy of the statements proved in the paper.