We apply the techniques of the Aubry-Mather theory for lattice systems to investigate the depinning transition of travelling waves for particle chains.

    Assume $B>0 $ is a critical value such that the particle chain has two homogeneous equilibria if the driving force $F/in [0,B]$. We demonstrate that there exist a transition threshold $0/leq F_c^+$ of the driving force such that the particle chain has stationary fronts for $0<F/leq F_c^+$ but no travelling fronts, and has travelling fronts but no stationary fronts if $F_c^+<F<B$. Moreover, we show that for $0<F<F_c^+$, besides stationary fronts, there are various kinds of equilibria such that the spatial shift has positive topological entropy on the set of equilibria. We also give a necessary and sufficient condition for $F_c^+=0$ via the existence of minimal foliation for the particle system without driving force, and prove that $F_c^+$ is continuous with respect to system parameters.

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