IEEE Circuits and Systems Magazine - Q3 2020 - 47

Abstract
Synchronization as a typical collective phenomenon emerges
in many complex systems, and has attracted world-wide attention from science, engineering, sociology, and so forth. With
the fast rising and development of network science, the Kuramoto-oscillator model is becoming one of the most pervasive
paradigms for investigating collective synchronous behaviors.
This review collects the latest advances of synchronization of
Kuramoto-oscillator networks as well as several applications,
such as locking of circuit oscillators, frequency synchronization in power grids, collective motion of self-propelled vehicles, and opinion synchronization in social networks. Besides,
we specially address the intra-layer and inter-layer synchronization of multi-layer Kuramoto-oscillator networks, along with
the stability analysis of Kuramoto-oscillator networkvs with
and without control. Finally, we conclude this review with the
discussions on some potential future directions for Kuramotooscillator networks.

T
	

1. Introduction
he past more than forty years have witnessed the
vigorous development of the celebrated Kuramoto-oscillator model [1], [2]:

io i (t ) = ~ i +

K
N

N

/ sin (i j (t) - i i (t)), i = 1, 2, f, N, (1)

j=1

where i i (t ) denotes the phase of the ith oscillator,
the derivative io i (t ) represents the instantaneous frequency, K is the overall coupling strength, and the intrinsic frequencies ~ i are distributed according to a
given probability density g (~) . On the one hand, by
the means of designing diverse network topologies,
varying the coupling strength, and selecting categories of distributions of initial phases and natural frequencies, the Kuramoto-oscillator networks present
distinct collective behaviors, such as progressive synchronization (from incoherence to synchrony) [1], [2],
chimera states [3], [4], explosive synchronization [5],
[6], cluster synchronization [7], and so forth. On the
other hand, the Kuramoto-oscillator model has been
applied into characterizing the dynamics in various
disciplines ranging from biology [8], chemistry [9], and
economics [10] to engineering [11]-[20], sociology [21],
seismology [22], etc. The Kuramoto - oscillator network, as a classical model for investigating the collective phenomena, still demonstrates its distinctive
fascination [23]-[27] up to now.
Digital Object Identifier 10.1109/MCAS.2020.3005485
Date of current version: 12 August 2020

The latest comprehensive overview [28] on synchronization in networks of Kuramoto oscillators covers
many relevant topics, mainly including the most used
analytical approaches, the analysis of numerical results, the impact of network patterns on the local and
global dynamics, and the developments on variations
of the Kuramoto model with the presence of noise and
inertia. Before that, Dörfler and Bullo in their survey
[29] have introduced several synchronization notions
and metrics, presented the sharpest known results,
and proposed analytical approaches for discussing
phase/frequency synchronization, partial synchronization, phase balancing, and pattern formation in
detail. Earlier review literatures [30], [31] have discussed the main available efforts at the beginning of
this century. With the rising of complex networks,
scholars break through the traditional single all-to-all
Kuramoto-oscillator network by endowing new structural configurations, which has been embraced by
early surveys [32], [33].
Nevertheless, with the discovery of new collective
phenomena and the development of network science,
control theory, and other disciplines, those aforementioned surveys [28]-[33] left some spaces and areas
that need to be supplemented and further improved. In
engineering applications, we focus on the locking of circuit oscillators, the frequency synchronization of power
grids, and the progress of collective motion of selfpropelled vehicles relating to the Kuramoto-oscillator
networks. Furthermore, keeping the eyes on the significance of cyber-physical-social interactions, this survey
reviews the application of Kuramoto-oscillator model in
social opinion synchronization.
In terms of the Kuramoto model itself, the critical
coupling strength [34], [35], as a key symbol of transition from incoherence to synchronization, and the novel adaptive couplings [36], [37] have been discussed in
detail. The requirements for achieving synchronization
in Kuramoto-oscillator networks have been addressed
among the coupling strength, the dispersion of initial
phases, and the distribution of natural frequencies
[38], [39]. In addition, the concept of multi-layer networks and network of networks has enriched the development of Kuramoto-type model in recent years, in
which many collective phenomena, such as chimera
states [40] and explosive synchronization [41]-[44],
have been also observed.
Apart from the collective phenomena observed in
numerical simulations, the rigorous stability analysis of

Jie Wu is with the Adaptive Networks and Control Lab, School of Information Science and Engineering, Fudan University, Shanghai, 200433, China, and
Xiang Li is with the Adaptive Networks and Control Lab, School of Information Science and Engineering, and the Research Center of Smart Networks
and Systems, School of Information Science and Engineering, Fudan University, Shanghai, 200433, China, (e-mail: lix@fudan.edu.cn).
THIRD QUARTER 2020 		

IEEE CIRCUITS AND SYSTEMS MAGAZINE	

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IEEE Circuits and Systems Magazine - Q3 2020

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