A stabilization problem for a class of nonlinear control systems is considered, Systems in this class can be viewed as a cascade connection of a linear time-invariant subsystem, a nonlinear time-periodic static subsystem, and an integrator. Hybrid logic-based feedback controllers are constructed to globally stabilize these systems to the origin. The controllers operate by switching between various time-periodic control functions at discrete-time instants, As specific applications, we consider stabilization of nonholonomic control systems in power form to the origin and stabilization of trajectories for a class of nonlinear control systems. Numerical examples of global stabilization and tracking are reported.