It is well known that extrema of classical, one-dimensional systems can be viewed as trajectories of a nonlinear, volume-preserving map. However, in general, thermodynamically stable extrema (i.e., local minima) of the free energy are numerically unstable trajectories of the map and so difficult to study by this technique. Here we explore a recent idea involving the use of a dissipative map to study the same kind of problem. Such a map may be designed so that its attractors are equal to, or close to, local minima of the energy. We apply such a dissipative map to the mean-field ANNNI model. We find that the technique reliably locates many kinds of metastable state: ferromagnetic, paramagnetic, commensurate, and incommensurate. However, the map also has two notable failings: it (apparently) has no chaotic attractors; and it fails to find correctly metastable states in a region of the phase diagram which is dominated by incommensurate states.