Poincaré half-maps can be used to characterize the behavior of recurrent dynamical systems. Their usefulness is demonstrated for a linear three-dimensional single-loop feedback system. In this example everything can be calculated analytically. The resulting half-maps are "benign" endomorphic maps with a complicated topological structure. This is surprising since the combination of two such half-maps (yielding an ordinary Poincaré map) always implies simple behavior in a linear system. The method has a direct bearing on the theory of piecewise linear systems -like the well-known Danziger-Elmergreen system of hormonal regulation.