Dynamics Of The Underdamped Frenkel-Kontorova Model

The Frenkel-Konotrova model is one of the most important model of nonlinear physics. Many physical phenomena and systems can be analyzed effectively with the help of Frenkel-Konotrova model and its generalizations. The equations of motion of the Frenkel-Konotrova model suject to damping and an external driving force are given by:where ε > 0, K > 0 and F(t) is periodic with period T = 2π/ω.Many studies of Frenkel-Konotrova model are based on the overdamped condition. However, in many numerical simulations, it was observed that the overdamped condition is sufficient but not necessary. In this article, we will investigate the dynamical behavior of the underdamped Frenkel-Konotrova model with large drift.We first study the Frenkel-Konotrova model with periodic boundary condition. We will prove that the system possesses an invariant restricted horizontal curve which is in fact an invariant circle if the phase space is regarded as a cylinder. We will also establish the relationship between the average velocity of the particles and the rotation number of the Poincare map on the invariant circle.In particular, we will study the Frenkel-Konotrova model without the coupling term, i.e., in the case of K = 0. At this time, the system is a single pendulum with damping and an external driving force. Besides the above results we show that the invariant curve is globally attracting.