Tracking unstable periodic orbits in nonstationary high-dimensional chaotic systems:Method and experiment

We consider the adaptive control of chaos in nonstationary high-dimensional dynamical systems. In particular, we propose and experimentally implement a technique to stabilize and track unstable periodic orbits based on the use of time series. In our technique, the position of the periodic orbit and other parameters in the controller are continually updated from recent measurements of the system state and perturbation histories, while the environment, simulated by one or several of the system's parameters, drifts independent of the control algorithm. We demonstrate the effectiveness of the technique computationally for the Hénon map, a chemical reaction model, and a coupled driven Duffing oscillator, and experimentally for a magnetoelastic ribbon system.