IEEE Circuits and Systems Magazine - Q3 2020 - 48

Kuramoto-oscillator networks has attracted more and
more attention in recent years [45]-[50]. Thereinto, a
great deal of sufficient conditions have been derived to
achieve phase agreement for the case of identical natural frequency and frequency synchronization for the
non-identical case, which usually relate to the network
topologies, coupling strength, initial phases, and other
factors in a network. In this way, the requirements for
ensuring synchronization have being relaxed continually by researchers, where the phase diameters are extended from a quarter of circle to the entire real space
[36], [37], [50]-[52], and the structure of networks is no
longer a complete and undirected graph. Furthermore,
the relatively weak bounded synchronization [53]-[58]
has been developed in recent years.
It is noteworthy that the prototypical and many other
modified Kuramoto-oscillator models can not arrive at
synchronization spontaneously in some detrimental
cases, such as the situations with quite small coupling
strengths, extremely dispersed initial phases and natural frequencies, very strong noise perturbations. Therefore, it is necessary to leverage external forces, e.g. controllers, to steer synchronization [59]-[61]. Nowadays,
many novel control strategies are proposed because
exerting external forces can bring some desired effects.
For example, introducing a pacemaker in networks can
drive all follower-oscillators to reach the anticipated
state [58], [62]-[67]. Besides, the multiplex control
protocols can further urge the Kuramoto-oscillator networks to achieve the finite-time and fixed-time synchronization [68], [69], where the upper bounds of synchronizing time can be estimated in advance. Even in noise
environments, the distributed control can guarantee the
global stochastic synchronization of Kuramoto-oscillator networks [70].
The rest of this overview is organized as follows. Section 2 presents several synchronization phenomena.
Section 3 collects several applications in engineering
and sociology. Then, Section 4 reports the recent development of synchronization of Kuramoto-oscillator

θj
r

ψ

Figure 1. Geometric interpretation of the order parameter [30].
48 	

model in multi-layer networks. The stability analysis of
Kuramoto-oscillator networks is elaborated in Section 5.
Finally, Section 6 concludes the review with the summary and discussions on some future directions.
2. Synchronization Phenomena
2.1. Progressive Synchronization
In Kuramoto's initial work, he assumes that the probability density g (~) of (1) is unimodal and symmetric
centered at its mean frequency ~ = ~r (e.g., Gaussian
distribution). Generally, we can set the mean frequency
to ~r = 0 by a shift, i.e., g (- ~) = g (~) . To quantify the
collective synchrony of the oscillator population distributed in the unit circle in the complex plane, Kuramoto introduces the order parameter
	

R (t ) = r (t ) e iW (t ) = 1
N

N

/ ei

i j (t )

, (2)

j=1

where the magnitude r (t ) ! [0, 1] of the order parameter
R(t) measures the coherence of the oscillator population, i is the imaginary unit, W(t ) is the average phase,
and the geometric interpretation is shown in Fig. 1.
There are two extreme cases: If the phases of all oscillators are spread in a complete incoherent state, then
r (t ) . 0; On the other hand, r (t) . 1 corresponds to
the case that all the oscillators move in a single tight
clump, which looks like a giant oscillator. Multiplying
both sides of Eq. (2) by e -iii (t) and equating the imaginary part yields
	

io i (t) = ~ i + Kr (t) sin ( W(t) - i i (t)), i = 1, 2, f, N. (3)

This formulation highlights the mean-field character
of the model, where each phase i i seems to evolve
independently from all the others, but couples to the
common average phase W(t) with coupling strength
K. Looking back on Eq. (1), the network topology of
Kuramoto-oscillators is all-to-all, and all phases interact with each other. In Eq. (3), as the increase of coupling strength K, each phase i i approaches to the mean
phase W(t), but not to any individual phase of oscillator. That is to say, the magnitude r(t) is proportional to
the coupling strength K with setting up a feedback relation between synchronization and coupling. More concretely, fixing g (~) to be a Gaussian or some other density with infinite tails, the macroscopic behavior of the
group of oscillators presents three states with respect
to coupling strength K (as shown in Fig. 2):
■■ Incoherent state. If the coupling strength K among
the oscillators is very small ( K 1 K c ), namely, the
interaction between each pair of oscillators is

IEEE CIRCUITS AND SYSTEMS MAGAZINE 		

THIRD QUARTER 2020
IEEE Circuits and Systems Magazine - Q3 2020

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