  We investigate the predictive power of recurrent neural networks for
oscillatory systems not only on the attractor, but in its vicinity as well. For
this we consider systems perturbed by an external force. This allows us to not
merely predict the time evolution of the system, but also study its dynamical
properties, such as bifurcations, dynamical response curves, characteristic
exponents etc. It is shown that they can be effectively estimated even in some
regions of the state space where no input data were given. We consider several
different oscillatory examples, including self-sustained, excitatory,
time-delay and chaotic systems. Furthermore, with a statistical analysis we
assess the amount of training data required for effective inference for two
common recurrent neural network cells, the long short-term memory and the gated
recurrent unit.
