  We explore the dynamics of strongly localized periodic solutions (discrete
solitons, or discrete breathers) in a finite one-dimensional chain of
asymmetric vibro-impact oscillators. The model involves a parabolic on-site
potential with asymmetric rigid constraints (the displacement domain of each
particle is finite), and a linear nearest-neighbor coupling. When the particle
approaches the constraint, it undergoes an impact (not necessarily elastic),
that satisfies Newton impact law. Nonlinearity of the system stems from the
impacts; their possible non-elasticity is the sole source of damping in the
system. We demonstrate that this vibro-impact model allows derivation of exact
analytic solutions for the asymmetric discrete breathers, both in conservative
and forced-damped settings. The asymmetry makes two types of breathers
possible: breathers that impact both or only one constraint. Transition between
these two types of the breathers corresponds to a grazing bifurcation. Special
character of the nonlinearity permits explicit derivation of a monodromy
matrix. Therefore, the stability of the obtained breather solutions can be
exactly studied in the framework of simple methods of linear algebra, and with
rather moderate computational efforts. All three generic scenarios of the loss
of stability (pitchfork, Neimark-Sacker and period doubling bifurcations) are
observed.
