A continuous stirred bioreactor with open-loop periodic behavior is analyzed under closed-loop conditions, using the dilution rate as control input. In the absence of control input bounds, the limiting substrate concentration converges to set point. When stringent bounds on control input are imposed, it becomes permanently saturated; then a limit cycle is generated, and the substrate concentration oscillates around an equilibrium point induced by saturation (EPIS). Suitable conditions are developed to avoid the presence of this closed-loop limit cycle. When the above conditions are satisfied, numerical simulations show that the substrate concentration in the closed-loop bioreactor has no permanent oscillations, the EPIS disappears and the limiting substrate concentration converges again to set point, in spite of the control input bounds.