In most of the networked dynamical system literature, the consensus and synchronization problems have been extensively studied. To illustrate the rich behaviors of networked dynamical systems, this chapter considers the existence of periodic behaviors for discrete-time second-order networked dynamical systems with input saturation constraints. We consider both cases where the agent dynamics are neutrally stable and double integrator, respectively. In both cases, we establish conditions on the feedback gains of the linear control law for achieving periodic behaviors. Simulation results are used to validate the theoretical results.