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We present the random behaviour of the Schr\"odinger map equation, ageometric partial differential equation, by considering its evolution forregular polygonal curves in both Euclidean and hyperbolic spaces. The resultsobtained are consistent with those for the vortex filament equation, anequivalent form of the Schr\"odinger map equation in the Euclidean space, andthus, provide a novel extension to its usefulness as a pseudorandom numbergenerator.

Pseudorandomness of the Schr\"odinger map equation