Synchronization and phase transition in the Frenkel-Kontorova-type model system

A coupled Hamiltonian system of the Frenkel-Kontorova model is studied in this paper, including the measure synchronization and the complete synchronization of the model. The measure synchronization depends not only on the coupling strength but also on the initial conditions. In a certain range of initial conditions, complete synchronization is reached when the system is in the measure synchronization, which means that as the coupling strength increases the energy can be transferred from one particle to another more easily. In other words, the average vibration amplitude of two particles tends to be identical as the coupling strength increases. The critical point Kc depends on both the initial total energy of two particles and the initial energy difference between two particles. Moreover, it is also related to the initial velocity directions of the two particles.