We study a model of coupled oscillators with bidirectional first nearest neighbors coupling with periodic boundary conditions. We show that a stable phase-locked solution is decided by the oscillators at the borders between the major clusters, which merge to form a larger one of all oscillators at the stage of complete synchronization. We are able to locate these four oscillators depending only on the set of the initial frequencies. Using these results plus an educated guess ͑supported by numerical findings͒ of the functional dependence of the corrections due to periodic boundary conditions, we are able to obtain a formula for the critical coupling, at which the complete synchronization state occurs. Such formula fits well in very good accuracy with the results that come from numerical simulations. This also helps to determine the sizes of the major clusters in the vicinity of the stage of full synchronization. © 2009 American Institute of Physics. ͓DOI: 10.1063/1.3212939͔Weakly coupled oscillators in the chaotic state have been known to represent many physical systems as well as chemical, biological, neurological, and so on. These systems synchronize in frequency under the influence of coupling. Knowing beforehand the value of the coupling constant and the dynamical behavior of the individual oscillators for complete synchronization to occur is an important source of information for real applications. This paper is a continuation of previous theoretical results for these systems. Here, we derive relationships that allow us to determine the oscillators which first lock in phase and drag the whole system into the synchronized state as well as the size of the two existing clusters before the transition.

We investigate synchronization in a Kuramoto-like model with nearest neighbour coupling. Upon analyzing the behaviour of individual oscillators at the onset of complete synchronization, we show that the time interval between bursts in the time dependence of the frequencies of the oscillators exhibits universal scaling and blows up at the critical coupling strength. We also bring out a key mechanism that leads to phase locking. Finally, we deduce forms for the phases and frequencies at the onset of complete synchronization.

A ballistic deposition model for the kinetics of surface growth for two species is introduced as a description of the evolution of a surface under vapor deposition. We used a tunable parameter P to control the deposition of the particles such that one type is deposited with probability P while the other is deposited with 1-P. Simulations in 2+1 dimensions using local surface diffusion lead to the formation of a rough surface whose dynamical evolution is not that of the Kardar-Parisi-Zhang universality class. Also, when surface diffusion becomes dominant, the model moves away from the Edwards-Wilkinson universality.