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We present an approximate dynamical systems model for the mass center trajectory of a tapped column of N uniform, inelastic, spheres (diameter d), in which collisional energy loss is governed by the Walton-Braun linear loading-unloading soft interaction. Rigorous analysis of the model, akin to the equations for the motion of a single bouncing ball on a vibrating plate, reveals a parameter γ:2aω2(1+e)/g that gauges the dynamical regimes and their transitions. In particular, we find bifurcations from periodic to chaotic dynamics that depend on frequency ω, amplitude a/d of the tap. Dynamics predicted by the model are also qualitatively observed in discrete element simulations carried out over a broad range of the tap parameters. [ABSTRACT FROM AUTHOR]