This paper studies a permutation binary neural network characterized by local binary connections, global permutation connections, and the signum activation function. Depending on the permutation connections, the network can generate various periodic orbits of binary vectors. Especially, we focus on globally stable periodic orbits such that almost all initial points fall into the orbits. In order to explore the periodic orbits, we present a simple evolutionary algorithm. Applying the algorithm to typical examples of PBNNs, existence of a variety of periodic orbits is clarified. Presenting an FPGA based hardware prototype, typical periodic orbits are confirmed experimentally. The hardware will be developed into various engineering applications such that stable control signals of switching circuits and stable approximation signals of time-series.