Network of Coupled Oscillators for Precision Timing (original) (raw)

Phase drift on networks of coupled crystal oscillators for precision timing

Physical Review E

Precise time dissemination and synchronization have been some of the most important technological tasks for several centuries. No later than Harrison's time, it was realized that precise time-keeping devices having the same stable frequency and precisely synchronized can have important applications in navigation. In modern times, satellite-based global positioning and navigation systems such as the GPS use the same principle. However, even the most sophisticated satellite navigation equipment cannot operate in every environment. In response to this need, we present a computational and analytical study of a network-based model of a high-precision, inexpensive, Coupled Crystal Oscillator System and Timing (CCOST) device. A bifurcation analysis (carried out by the authors in a related publication) of the network dynamics shows a wide variety of collective patterns, mainly various forms of discrete rotating waves and synchronization patterns. Results from computer simulations seem to indicate that, among all patterns, the standard traveling wave pattern in which consecutive crystals oscillate out of phase by 2π/N , where N is the network size, leads to phase drift error that decreases as 1/N as opposed to 1/ √ N for an uncoupled ensemble. The results should provide guidelines for future experiments, design and fabrication tasks.

Network Synchronization Revisited: Time Delays in Mutually Coupled Oscillators

IEEE Access, 2022

Coordinated and efficient operation in large, complex systems requires the synchronization of the rhythms of spatially distributed components. Such systems are the basis for critical infrastructure such as satellite navigation, mobile communications, and services like the precision time protocol and Universal Coordinated Time. Different concepts for the synchronization of oscillator networks have been proposed, in particular mutual synchronization without and hierarchical synchronization from a reference clock. Established network synchronization models in electrical engineering address the role of inevitable cross-coupling time delays for network synchronization. Mutual synchronization has been studied using linear approximations of the coupling functions of these models. We review previous work and present a general model in which we study synchronization taking into account nonlinearities and finite time delays. As a result, dynamical phenomena in networks of coupled electronic oscillators induced by time delays, such as the multistability and stabilization of synchronized states can be predicted and observed. We study the linear stability of nonlinear states and predict for which system parameters synchronized states can be stable. We use these results to discuss the implementation of mutual synchronization for complex system architectures. A key finding is that mutual synchronization can result in stable in-and anti-phase synchronized states in the presence of large time delays. We provide the condition for which such synchronized states are guaranteed to be stable. INDEX TERMS Synchronization, delay effects, systems engineering and theory, control theory, phase locked loops, mutual coupling.

Synchronizing large number of nonidentical oscillators with small coupling

EPL (Europhysics Letters), 2012

The topic of synchronization of oscillators has attracted great and persistent interest, and all previous conclusions and intuitions have convinced that large coupling is required for synchronizing a large number of coupled nonidentical oscillators. Here the influences of different spatial frequency distributions on the efficiency of frequency synchronization are investigated by studying arrays of coupled oscillators with diverse natural frequency distributions. A universal log-normal distribution of critical coupling strength Kc for synchronization irrespective of the initial natural frequency is found. In particular, a physical quantity "roughness " R of spatial frequency configuration is defined, and it is found that the efficiency of synchronization increases monotonously with R. For large R we can reach full synchronization of arrays with a large number of oscillators at finite Kc. Two typical kinds of synchronization, the "multiple-clustering" one and the "single-center-clustering" one, are identified for small and large R's, respectively. The mechanism of the latter type is the key reason for synchronizing long arrays with finite Kc.

Oscillator Networks: Delays and Synchronization

IFAC-PapersOnLine, 2021

The present paper addresses the relationships between self-oscillations, synchronization and time-delays within oscillator networks. It relies on the ideas going from J.L. Lagrange to J.K. Hale and suggests some open problems to future research. Shortly, an oscillator is defined as a dynamical structure displaying a transient process called oscillation. An oscillator network can be defined as a lattice of oscillators. If this lattice structure is “endowed” with an attractor, then one can conceive the idea of synchronization: simpler the attractor structure is, higher is the degree of synchronization. The paper focuses on Huygens synchronization consisting of synchronizing two oscillators through a 1D distributed medium (rope, string, rod, LC transmission line) inducing propagation delays. The considered oscillators are Van der Pol oscillators (mechanical or electrical) and synchronization takes place through local oscillations quenching (called by the physicists - radiation dissipat...

Average distance as a predictor of synchronisability in networks of coupled oscillators

2010

Abstract The importance of networks of coupled oscillators is widely recognized. Such networks occur in biological systems like the heart, in chemical systems, in computational problems, and in engineering systems. Systems of coupled oscillators can also be used as an abstract model for synchronisation in organisations. Here we show that synchronisability in a specific coupledoscillator model, the Kuramoto model, is best predicted using the average distance (or characteristic path length) between nodes in the network.

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.

Mechanisms of synchronization and pattern formation in a lattice of pulse-coupled oscillators

Physical Review E, 1998

Among the collective phenomena that are currently attracting the interest of the scientific community one of the most relevant concerns the synchronization of the temporal activity of populations of interacting nonlinear oscillators, due to its ubiquity in many different fields of science. Experimental evidences of this phenomenon have been reported for centuries 1 but in the last decades the advance in the comprehension of its nature has allowed the development of a theoretical description. In this context, several successful ideas has ...

Synchronization in complex networks of phase oscillators: A survey

Automatica, 2014

The emergence of synchronization in a network of coupled oscillators is a fascinating subject of multidisciplinary research. This survey reviews the vast literature on the theory and the applications of complex oscillator networks. We focus on phase oscillator models that are widespread in real-world synchronization phenomena, that generalize the celebrated Kuramoto model, and that feature a rich phenomenology. We review the history and the countless applications of this model throughout science and engineering. We justify the importance of the widespread coupled oscillator model as a locally canonical model and describe some selected applications relevant to control scientists, including vehicle coordination, electric power networks, and clock synchronization. We introduce the reader to several synchronization notions and performance estimates. We propose analysis approaches to phase and frequency synchronization, phase balancing, pattern formation, and partial synchronization. We present the sharpest known results about synchronization in networks of homogeneous and heterogeneous oscillators, with complete or sparse interconnection topologies, and in finite-dimensional and infinite-dimensional settings. We conclude by summarizing the limitations of existing analysis methods and by highlighting some directions for future research.

Enhancing synchronization in complex networks of coupled phase oscillators

2007

By a model of coupled phase oscillators, we show analytically how synchronization in non-identical complex networks can be enhanced by introducing a proper gradient into the couplings. It is found that, by pointing the gradient from the large-degree to the small-degree nodes on each link, increase of the gradient strength will bring forward the onset of network synchronization monotonically, and, with the same gradient strength, heterogeneous networks are more synchronizable than homogeneous networks. The findings are tested by extensive simulations and good agreement are found.

Synchronization of Phase-coupled Oscillators with Distance-dependent Delay

Eprint Arxiv 1008 0494, 2010

By means of numerical integration we investigate the coherent and incoherent phases in a generalized Kuramoto model of phase-coupled oscillators with distance-dependent delay. Preserving the topology of a complete graph, we arrange the nodes on a square lattice while introducing finite interaction velocity, which gives rise to non-uniform delay. It is found that such delay facilitates incoherence and removes reentrant behavior found in models with uniform delay. A coupling-delay phase diagram is obtained and compared with previous results for uniform delay.