Rotation number of the overdamped Frenkel-Kontorova model with ac-driving

The dynamical behavior of the overdamped and overdamped limit cases of the Frenkel-Kontorova model with ac-driving is studied. The relationship is established between the average velocity of the particles and the rotation number of the orientation preserving homeomorphism on the circle induced from the Poincaré map. The dynamical behavior for both cases is determined completely by the rotation number. Numerical results are also provided for the second order system to support the theoretical conclusions.