Determination of the critical coupling for oscillators in a ring

Abstract

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.

Multistable behavior above synchronization in a locally coupled Kuramoto model

“…Nevertheless there is a sine equal to ±1 near the bifurcation, with the difference becoming smaller as N → ∞ in agreement with previous literature where this fact has played a crucial role [15,16]. In the next section we will show how these deviations and the multiple solutions are generated from a chain of oscillators.…”

“…The ring topology is defined by the periodic conditions θ N+1 = θ 1 and θ 0 = θ N . There is a minimum value for the coupling constant K, denoted as critical synchronization coupling K s , that drives the system into a fully synchronized state [16][17][18]. In this state the oscillators' instantaneous frequencies assume a constant value = 1…”

“…At this moment the frequency becomes constant and the phases lock, such that all phase differences are constant. The solution for full synchronization and its stability has been studied by many authors [15][16][17][18][19]. Zheng et al [12] already in 1998 pointed out that the behavior of the order parameter "indicates the coexistence of multiple attractors of phase locking states" above the synchronization critical coupling, K s .…”

Synchronization of a forced self-sustained Duffing oscillator

“…(7) are δA and φ s , while δΩ is given in terms of the parameters of our problem through the first of Eqs. (6). A representation of Eq.…”

“…Mathematical, computational, and experimental models have helped to detect and understand the common elementary mechanisms that drive synchronization in many of those systems. Abstract models of coupled oscillators have become a very fruitful tool for the analytical study of coherent evolution in Nature [4,5,6,7].…”