  A discrete and periodic complex Ginzburg-Landau equation, coupled to a
discrete mean equation, is systematically derived from a driven and dissipative
oscillator model, close to the onset of a supercritical Hopf bifurcation. The
oscillator model is inspired by recent experiments exploring active vibrations
of quasi-one-dimensional lattices of self-propelled millimetric droplets
bouncing on a vertically vibrating fluid bath. Our systematic derivation
provides a direct link between the constitutive properties of the lattice
system and the coefficients of the resultant amplitude equations, paving the
way to compare the emergent nonlinear dynamics---namely discrete bright and
dark solitons, breathers, and traveling waves---against experiments. Further,
the amplitude equations allow us to rationalize the successive bifurcations
leading to these distinct dynamical states. The framework presented herein is
expected to be applicable to a wider class of oscillators characterized by the
presence of a dynamic coupling potential between particles. More broadly, our
results point to deeper connections between nonlinear oscillators and the
physics of active and driven matter.
