  We consider a massive particle driven with a constant force in a periodic
potential and subjected to a dissipative friction. As a function of the drive
and damping, the phase diagram of this paradigmatic model is well known to
present a pinned, a sliding, and a bistable regime separated by three distinct
bifurcation lines. In physical terms, the average velocity $v$ of the particle
is nonzero only if either (i) the driving force is large enough to remove any
stable point, forcing the particle to slide, or (ii) there are local minima but
the damping is small enough, below a critical damping, for the inertia to allow
the particle to cross barriers and follow a limit cycle; this regime is
bistable and whether $v > 0$ or $v = 0$ depends on the initial state. In this
paper, we focus on the asymptotes of the critical line separating the bistable
and the pinned regimes. First, we study its behavior near the "triple point"
where the pinned, the bistable, and the sliding dynamical regimes meet. Just
below the critical damping we uncover a critical regime, where the line
approaches the triple point following a power-law behavior. We show that its
exponent is controlled by the normal form of the tilted potential close to its
critical force. Second, in the opposite regime of very low damping, we revisit
existing results by providing a simple method to determine analytically the
exact behavior of the line in the case of a generic potential. The analytical
estimates, accurately confirmed numerically, are obtained by exploiting exact
soliton solutions describing the orbit in a modified tilted potential which can
be mapped to the original tilted washboard potential. Our methods and results
are particularly useful for an accurate description of underdamped nonuniform
oscillators driven near their triple point.
