  We study the dynamics of a piecewise-linear second-order delay differential
equation that is representative of feedback systems with relays (switches) that
actuate after a fixed delay. The system under study exhibits strong
multirhythmicity, the coexistence of many stable periodic solutions for the
same values of the parameters. We present a detailed study of these periodic
solutions and their bifurcations. Starting from an integro-differential model,
we show how to reduce the system to a set of finite-dimensional maps. We then
demonstrate that the parameter regions of existence of periodic solutions can
be understood in terms of discontinuity induced bifurcations and their
stability is determined by smooth bifurcations. Using this technique we are
able to show that slowly oscillating solutions are always stable if they exist.
We also demonstrate the coexistence of stable periodic solutions with
quasiperiodic solutions.
