Phase transitions on an array of double-threshold noisy devices with a positive feedback

We consider solutions of the self-consistent equation of an array of double threshold noisy devices with a positive feedback of an aggregation of the output from each device. Stationary solutions of the aggregation (order parameter) as a function of the input signal are investigated. We classify stationary solutions as three types: (1) monotonically increasing function (for large noise strength and small feedback strength), which is approximated by a linear function around a small input signal; (2) single-hysteresis function (for a large feedback strength), which is observed in the Ising model; (3) double-hysteresis function (for a small noise strength and a small feedback strength), which is never found in the Ising model. The diagram to show the three different types is theoretically calculated. We discuss a method to improve a performance of a stochastic resonator by tuning a system near a phase transition point. We conclude that the response of the system exhibits three different types of behaviors depending on the feedback strength and the standard deviation of the noise