Strong chaos without the butterfly effect in dynamical systems with feedback

We discuss the predictability of a system that drives a chaotic system with a positive Lyapunov exponent. In the absence of feedback, the driver is regular and fully predictable. With a small feedback of strength epsilon, the state of the driver can be predicted up to a time diverging with a power of epsilon(-1), although the total system is strongly chaotic. The exponential amplification of the uncertainty on the initial conditions of the driver coexists with very long predictability times as illustrated in a model of coupled maps and of three point vortices in a disc.