  We present the random behaviour of the Schr\"odinger map equation, a
geometric partial differential equation, by considering its evolution for
regular polygonal curves in both Euclidean and hyperbolic spaces. The results
obtained are consistent with those for the vortex filament equation, an
equivalent form of the Schr\"odinger map equation in the Euclidean space, and
thus, provide a novel extension to its usefulness as a pseudorandom number
generator.
