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The main purpose is to describe the evolution of $\Xt = \Xs \wedge_- \Xss,$with $\X(s,0)$ a regular polygonal curve with a nonzero torsion in the3-dimensional hyperbolic space. Unlike in the Euclidean space, a nonzerotorsion implies two different helical curves. However, recent techniquesdeveloped by de la Hoz, Kumar, and Vega help us in describing the evolution atrational times both theoretically and numerically, and thus, the similaritiesand differences.  Numerical experiments show that the trajectory of the point $\X(0,t)$exhibits new variants of Riemann's non-differentiable function whose structuredepends on the initial torsion in the problem. As a result, with these newsolutions, it is shown that the smooth solutions (helices, straight line) inthe hyperbolic space show the same instability as displayed by their Euclideancounterparts and curves with zero-torsion. These numerical observations are inagreement with some recent theoretical results obtained by Banica and Vega.

On the Schr\"{o}dinger map for regular helical polygons in the hyperbolic space