  Piecewise linear recurrent neural networks (PLRNNs) form the basis of many
successful machine learning applications for time series prediction and
dynamical systems identification, but rigorous mathematical analysis of their
dynamics and properties is lagging behind. Here we contribute to this topic by
investigating the existence of n-cycles $(n\geq 3)$ and border-collision
bifurcations in a class of n-dimensional piecewise linear continuous maps which
have the general form of a PLRNN. This is particularly important as for
one-dimensional maps the existence of 3-cycles implies chaos. It is shown that
these n-cycles collide with the switching boundary in a border-collision
bifurcation, and parametric regions for the existence of both stable and
unstable n-cycles and border-collision bifurcations will be derived
theoretically. We then discuss how our results can be extended and applied to
PLRNNs. Finally, numerical simulations demonstrate the implementation of our
results and are found to be in good agreement with the theoretical derivations.
Our findings thus provide a basis for understanding periodic behavior in
PLRNNs, how it emerges in bifurcations, and how it may lead into chaos.
