  Reservoir computing is a bio-inspired computing paradigm for processing
time-dependent signals. Its hardware implementations have received much
attention because of their simplicity and remarkable performance on a series of
benchmark tasks. In previous experiments the output was uncoupled from the
system and in most cases simply computed offline on a post-processing computer.
However, numerical investigations have shown that feeding the output back into
the reservoir would open the possibility of long-horizon time series
forecasting. Here we present a photonic reservoir computer with output
feedback, and demonstrate its capacity to generate periodic time series and to
emulate chaotic systems. We study in detail the effect of experimental noise on
system performance. In the case of chaotic systems, this leads us to introduce
several metrics, based on standard signal processing techniques, to evaluate
the quality of the emulation. Our work significantly enlarges the range of
tasks that can be solved by hardware reservoir computers, and therefore the
range of applications they could potentially tackle. It also raises novel
questions in nonlinear dynamics and chaos theory.
