A novel preserved partial order for cooperative networks of units with overdamped second order dynamics, and application to tilted Frenkel-Kontorova chains

We introduce a novel partial order on the state space for cooperative networks of units with overdamped second order dynamics, which is preserved under the forward dynamics. Among other things, we use it to prove that for chains of particles in a tilted periodic potential with overdamped inertial dynamics, for each class of spatially periodic states if there are no rotationally ordered equilibria then there is a globally attracting periodically sliding solution.