It is known that for single-input neutrally stable planar systems, there exists a class of saturated globally stabilizing linear state feedback control laws. The goal of this paper is to characterize the dynamic behavior for such a system under arbitrary locally stabilizing linear state feedback control laws. On the one hand, for the continuous-time case, we show that all locally stabilizing linear state feedback control laws are also globally stabilizing control laws. On the other hand, for the discrete-time case, we first show that this property does not hold by explicitly constructing nontrivial periodic solution for a particular system. We then show for an example that there exists more globally stabilizing linear state feedback control laws than well known ones in the literature.