Mean motion resonances (MMR) are a frequent phenomenon among extrasolar planetary systems. Current observations indicate that many systems have planets that are close to or inside the 2:1 MMR, when the orbital period of one of the planets is twice the other. Analytical models to describe this particular MMR can only be reduced to integrable approximations in a few specific cases. While there are successful approaches to the study of this MMR in the case of very elliptic and/or very inclined orbits using semi-analytical or semi-numerical methods, these may not be enough to completely understand the resonant dynamics. In this work, we propose to apply a well-established numerical method to assess the global portrait of the resonant dynamics, which consists in constructing dynamical maps. Combining these maps with the results from a semi-analytical method, helps to better understand the underlying dynamics of the 2:1 MMR, and to identify the behaviors that can be expected in different regions of the phase space and for different values of the model parameters. We verify that the family of stable resonant equilibria bifurcate from symmetric to asymmetric librations, depending on the mass ratio and eccentricities of the resonant planets pair. This introduces new structures in the phase space, that turns the classical V-shape of the MMR, in the semi-major axis vs. eccentricity space, into a sand clock shape. We construct dynamical maps for three extrasolar planetary systems, TOI-216, HD27894, and K2-24, and discuss their phase space structure and their stability in the light of the orbital fits available in the literature.