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 resea rch. In this paper, we propose an adaptive variable structure time-delayed self-control design using only partial information of states for tracking i nherent unstable periodic orbits (UPO's) in chaotic dynamic systems. Since the period of inherent UPO's is usually difficult to obtain, a gradient-des cent-based adaptive search algorithm for the time-delay constant is utilize d. A variable structure control (VSC) mechanism is employed to create an at traction region about the UFO such that once the trajectory enters the regi on, it will stay in it forever. Due to the ergodicity of chaotic dynamics, such an attraction region is always reachable. Two well-known chaotic dynam ics, the Duffing equation and the Lorenz system, are used to demonstrate th e effectiveness of the proposed approach.