In this talk, we study elliptic helicoidal translating solitons of the mean curvature flow in three-dimensional Minkowski space, that is, helicoidal translators whose axis is timelike. By expressing the generating curve in terms of its Frenet frame, we reformulate the translator equation in terms of curvature and support functions, obtaining a planar system of ordinary differential equations. We then employ techniques from the qualitative theory of dynamical systems, together with an inversion that transforms the dynamics at infinity into a bounded region, to investigate the equilibrium points and the global behavior of the phase portrait. A bifurcation occurs at the critical pitch value \(h=2\), where the number of equilibrium points increases from two to six. This analysis leads to a complete classification of the corresponding translating solitons and a description of their causal characters. Our results illustrate how dynamical systems techniques provide an effective framework for understanding the geometry and global structure of helicoidal translators in Lorentzian settings.