Geometrical properties of coupled oscillators at synchronization (original) (raw)
Related papers
Synchronization in a chain of nearest neighbors coupled oscillators with fixed ends
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2003
We investigate a system of coupled phase oscillators with nearest neighbors coupling in a chain with fixed ends. We find that the system synchronizes to a common value of the time-averaged frequency, which depends on the initial phases of the oscillators at the ends of the chain. This time-averaged frequency decays as the coupling strength increases. Near the transition to the frozen state, the time-averaged frequency has a power law behavior as a function of the coupling strength, with synchronized time-averaged frequency equal to zero. Associated with this power law, there is an increase in phases of each oscillator with 2 jumps with a scaling law of the elapsed time between jumps. During the interval between the full frequency synchronization and the transition to the frozen state, the maximum Lyapunov exponent indicates quasiperiodicity. Time series analysis of the oscillators frequency shows this quasiperiodicity, as the coupling strength increases.
Determination of the critical coupling for oscillators in a ring
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2009
We study a model of coupled oscillators with bidirectional first nearest neighbours 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 as well as the size of major clusters in the vicinity of the stage of full synchronization which we show to depend only on the set of initial frequencies. Using the method presented here, we are able to obtain an analytic form of the critical coupling, at which the complete synchronization state occurs. PACS numbers: 05.45.Xt, 05.45.-a, 05.45.Jn
Transition to complete synchronization in phase-coupled oscillators with nearest neighbor coupling
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2009
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. PACS numbers: 05.45.Xt, 05.45.-a, 05.45.Jn
Cluster synchrony in systems of coupled phase oscillators with higher-order coupling
Physical Review E, 2011
We study the phenomenon of cluster synchrony that occurs in ensembles of coupled phase oscillators when higher-order modes dominate the coupling between oscillators. For the first time, we develop a complete analytic description of the dynamics in the limit of a large number of oscillators and use it to quantify the degree of cluster synchrony, cluster asymmetry, and switching. We use a variation of the recent dimensionality-reduction technique of Ott and Antonsen (Chaos 18, 037113 (2008)) and find an analytic description of the degree of cluster synchrony valid on a globally attracting manifold. Shaped by this manifold, there is an infinite family of steady-state distributions of oscillators, resulting in a high degree of multi-stability in the cluster asymmetry. We also show how through external forcing the degree of asymmetry can be controlled, and suggest that systems displaying cluster synchrony can be used to encode and store data.
Computational explorations into the dynamics of rings of coupled oscillators
Applied Mathematics and Computation, 2002
This paper deals with the response of homogeneous and inhomogeneous rings of coupled oscillators where each individual oscillator, when uncoupled from the others, is chaotic. It is shown that coupling can bring about a wide variety of global responses, and that there is a significant range of coupling values when the response of the ring is periodic despite the fact that each oscillator is chaotic. In fact numerous periodic solutions can be found depending on the initial conditions. The response of a coupled set of homogeneous and non-homogeneous rings is also investigated showing that the behavior of such coupled compartmental models can be quite counterintuitive and sensitive to the parameters that describe the extent and nature of coupling. Ó
Synchronization of coupled oscillators
In this work we begin by introducing the Kuramoto model, constructing its solutions in the thermodynamic limit and showing the close connection between statistical physics and dynamical systems that lead to the main theoretical insights. The systematic study of a finite population of self sustained oscillators began in the first decade of this century. Unlike most of the papers we have found, we are not interested in the synchronization transition in itself but rather in phase locked patterns and their relation with frequency distribution among oscillators. The problem of stability, as we have already mentioned, experienced great advances in recent years. In a brief discussion we only address the problem of stability of the simplest solution allowed by the Kuramoto model: the incoherent solution. After that we introduce Chimera states, First noticed by Kuramoto and his colleagues in which the introduction of a non local coupling gives origin to a split in a region with synchronised oscillators and other with asynchronous one. Then we proceed by exploring the literature and the results with a fnite number of oscillators, model explored with persistence only since mainly 2004. But here we are yet in Kuramoto framework which is abandoned, in a rigorous terminology, when we pursuit structured and not all-to-all coupling. Although we could introduce the same mean models quantities if well defined in each situation, this did not help us in making sense of the results and is not an help in any analytical work. In our analysis of a ring of coupled oscillators we construct a space that allows us to relate the stable solutions with the eigenvectors of the laplacian of the graph in which we work. work.
Single-clustering synchronization in a ring of Kuramoto oscillators
Journal of Physics A: Mathematical and Theoretical, 2014
Multiple-clustering synchronization is a common scenario of global phase synchronization. However, a novel single-clustering type of phase synchronization in a ring of Kuramoto oscillators has been recently reported in studying the influence of the permutation of the natural frequencies of oscillators on the synchronization efficiency (Wu et al 2012 Europhys. Lett. 97 40005). It was found that it occurs for a particular spatial frequency distribution and gives rise to a very small critical coupling strength K c even if the oscillator number is large. Here we focus on this particular type of synchronization and study its generality. We provide some solid evidence for the convergence of K c to a small constant in the thermodynamic limit, based on the finite size analysis. Further we demonstrate that it is robust in the sense of either switching the natural frequencies of any two oscillators or randomly perturbing the frequencies of all coupled oscillators. All these findings prove that the single-clustering synchronization is indeed generically observable, with merit for potential engineering applications.
Transition to complete synchronization in phase coupled oscillators with nearest neighbours coupling
2008
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.
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2013
We experimentally study the complex dynamics of a unidirectionally coupled ring of four identical optoelectronic oscillators. The coupling between these systems is time-delayed in the experiment and can be varied over a wide range of delays. We observe that as the coupling delay is varied, the system may show different synchronization states, including complete isochronal synchrony, cluster synchrony, and two splay-phase states. We analyze the stability of these solutions through a master stability function approach, which we show can be effectively applied to all the different states observed in the experiment. Our analysis supports the experimentally observed multistability in the system. V C 2013 AIP Publishing LLC. [http://dx.