  We investigate in detail the asymptotic properties of tachyon cosmology for a
broad class of self-interaction potentials. The present approach relies in an
appropriate re-definition of the tachyon field, which, in conjunction with a
method formerly applied in the bibliography in a different context, allows to
generalize the dynamical systems study of tachyon cosmology to a wider class of
self-interaction potentials beyond the (inverse) square-law one. It is revealed
that independent of the functional form of the potential, the matter-dominated
solution and the ultra-relativistic (also matter-dominated) solution, are
always associated with equilibrium points in the phase space of the tachyon
models. The latter is always the past attractor, while the former is a saddle
critical point. For inverse power-law potentials $V\propto\phi^{-2\lambda}$ the
late-time attractor is always the de Sitter solution, while for sinh-like
potentials $V\propto\sinh^{-\alpha}(\lambda\phi)$, depending on the region of
parameter space, the late-time attractor can be either the inflationary
tachyon-dominated solution or the matter-scaling (also inflationary) phase. In
general, for most part of known quintessential potentials, the late-time
dynamics will be associated either with de Sitter inflation, or with
matter-scaling, or with scalar field-dominated solutions.
