Controlling chaos for nonautonomous systems by detecting unstable periodic orbits

Designing the Ott–Grebogi–Yorke controller for controlling a nonautonomous chaotic system with forcing input requires detecting the unstable periodic orbits (UPOs) embedded in the chaotic motion. A simple detection approach based on the shooting method is presented in this study. This improvement of the close returns (CR) method can find multiple unstable periodic orbits of the same period and unstable periodic orbits of long periods by a short time series. Thus the UPOs are quickly determined. For real applications, system models are unknown. To obtain mathematical model, this work applies a nonlinear identification method, based on the harmonic balance principle, to a chaotic unknown system with a noisy environment. A Duffing equation is presented as a numerical example in this study. To simulate experimental conditions, random data are added to a sampled time series. Furthermore, the OGY controller is designed based on the data evaluated by the shooting method. The simulation results demonstrate that the proposed method is both accurate and feasible.