Existence of dynamical hull functions with two variables for the ac-driven Frenkel–Kontorova model

In this paper we are concerned with the dynamics of incommensurate structures in the overdamped limit case of the Frenkel–Kontorova model with periodic driving force. We show in the infinite-dimensional phase space, there exists an ordered circle, on which both the Poincaré map and the space shift map induce orientation preserving circle homeomorphisms with rotation numbers ρ and ω   respectively, the latter characterizing the mean spacing of particles. Furthermore, we prove that the average velocity ν¯ for the particle chain with irrational ω   exists and ν¯=ρ/T, where T denotes the period of the driving force, and it is continuous and nondecreasing with respect to F, the average of the external driving force. If the shift map on the invariant ordered circle is ergodic, we demonstrate the existence of two-variable dynamical hull functions and that there are no mode locking behaviors.