  While considerable progress has been made in the analysis of large systems
containing a single type of coupled dynamical component (e.g., coupled
oscillators or coupled switches), systems containing diverse components (e.g.,
both oscillators and switches) have received much less attention. We analyze
large, hybrid systems of interconnected Kuramoto oscillators and Hopfield
switches with positive feedback. In this system, oscillator synchronization
promotes switches to turn on. In turn, when switches turn on they enhance the
synchrony of the oscillators to which they are coupled. Depending on the choice
of parameters, we find theoretically coexisting stable solutions with either
(i) incoherent oscillators and all switches permanently off, (ii) synchronized
oscillators and all switches permanently on, or (iii) synchronized oscillators
and switches that periodically alternate between the on and off states.
Numerical experiments confirm these predictions. We discuss how transitions
between these steady state solutions can be onset deterministically through
dynamic bifurcations or spontaneously due to finite-size fluctuations.
