  This work focuses on the study of time-periodic solutions, including
breathers, in a nonlinear lattice consisting of elements whose contacts
alternate between strain-hardening and strain-softening. The existence,
stability, and bifurcation structure of such solutions, as well as the system
dynamics in the presence of damping and driving are studied systematically. It
is found that the linear resonant peaks in the system bend toward the frequency
gap in the presence of nonlinearity. The time-periodic solutions that lie
within the frequency gap compare well to Hamiltonian breathers if the damping
and driving are small. In the Hamiltonian limit of the problem, we use a
multiple scale analysis to derive a Nonlinear Schr\"odinger (NLS) equation to
construct both acoustic and optical breathers. The latter compare very well
with the numerically obtained breathers in the Hamiltonian limit.
