Tracking inherent periodic orbits in chaotic dynamic systems via adaptive variable structure time-delayed self control

Tracking inherent periodic orbits is of significance in chaos control research. In this paper, we propose an adaptive variable structure time-delayed self-control design using only partial information of states for tracking inherent unstable periodic orbits (UPO's) in chaotic dynamic systems. Since the period of inherent UPO's is usually difficult to obtain, a gradient-descent-based adaptive search algorithm for the time-delay constant is utilized. A variable structure control (VSC) mechanism is employed to create an attraction region about the UPO such that once the trajectory enters the region, it will stay in it forever. Due to the ergodicity of chaotic dynamics, such an attraction region is always reachable. Two well-known chaotic dynamics, the Duffing equation and the Lorenz system, are used to demonstrate the effectiveness of the proposed approach.