Robust and efficient interaction with complex systems

Low-dimensional chaotic map dynamics has been successfully used to predict the dynamics of high-dimensional systems far from equilibrium. For control, low-dimensional models can be used only if the control force is very small, otherwise hidden degrees of freedom may become excited. We study the control of chaotic map dynamics with extremely small forcing functions. We find that the smallest forcing function, which is called a resonant forcing function, echoes the natural dynamics of the system. This means, when the natural dynamics is irregular, the optimal forcing function is irregular too. If the natural dynamics contains a certain periodicity, the optimal forcing function contains that periodicity too. We show that such controls are effective even if the system has hidden degrees of freedom and if the probes of the control system have a low resolution. Further we show that resonant forcing functions of chaotic systems decrease exponentially, where the rate equals the Liapunov exponent of the unperturbed system. We apply resonant forcing functions for efficient control of chaotic systems and for system identification.