Stabilization of stochastic cycles and chaos suppression for nonlinear discrete-time systems

In this paper we consider a nonlinear discrete-time control system with regular and chaotic dynamics forced by stochastic disturbances. The problem addressed is the design of the feedback regulator which stabilizes a limit cycle of the closed-loop deterministic system and synthesizes a required dispersion of random states for the corresponding stochastic system. To solve this problem, we propose a new method based on the stochastic sensitivity function technique. This function approximates a dispersion of random states distributed around deterministic cycle. Explicit formulas for the intercoupling between stochastic sensitivity function and considered system parameters are worked out. The problem of the design of the required stochastic sensitivity function for cycles by feedback regulators is solved. Coefficients of the feedback regulator are constructed and corresponding attainability sets are described. The effectiveness of the proposed approach is demonstrated on the stochastic Verhulst model. It is shown that constructed regulators provide a low level of sensitivity and suppress chaotic oscillations.