Coupled first-order digital phase-locked loops (DPLLs), have been proposed in digital communications and synchronization applications. Previously, coupled DPLLs have been studied via extensive computer simulations. We obtain dynamics for a generalized class of oscillators, including the DPLLs. The dynamics live on two 'pinched annuli', continuous everywhere but at the pinches. We prove results concerning the existence of all fixed points and period-two points for the general dynamics, briefly discuss the structural stability of these orbits, and then apply our results to the coupled DPLLs. We also find compelling numerical evidence of a horseshoe (specifically, a subshift of finite-type) in the coupled DPLL dynamics; a proof of this horseshoe is forthcoming.