Reduction Of A Class Of Coupled Oscillators Systems And Stability Of Travelling Wave Solutions In The Underdamped F-K Model

In this paper we discuss the dynamical behavior of a class of coupled oscillators with periodic on-site potential and convex interaction potential. We prove by monotonicity approach that the Poincarémap of the systems with periodic boundary conditions or Neumann boundary conditions admits an invariant curve, on which the Poincarémap is actually an orientation preserving circle homeomorphism. Then some consequences are discussed, including the existence and uniqueness of the average velocity, the existence of running periodic solutions, the occurrence of frequency synchronization, and a Massera type theorem. Under Dirichlet boundary conditions, we study the boundness of the solutions. The conclusions for the cooperative systems could be extended to the competitive systems by making a transformation of time scale.We also show by the strong monotonicity that the travelling wave solution for the underdamped Frenkel-Kontorova model with dc-driving and periodic boundary conditions is globally stable provided the driving force is large enough.