IEEE Circuits and Systems Magazine - Q3 2020 - 50

Another notion is weak chimera which has been defined as a type of the invariant set showing partial frequency synchronization [82]. Besides the chimera states
are typically observed in large ensembles of oscillators
and analyzed in the continuum limit, these states may
also occur in systems with finite (and small) numbers of
oscillators [83]. By taking the inertia into consideration,
chimera states can occur in second-order coupled Kuramoto-oscillator networks [84]. Recently, Bera et al. have
investigated the effect of different coupling topologies
on chimera states [85]. Furthermore, Laing has considered the situation of heterogeneous networks, in which
the natural frequencies of oscillators are nonidentical
[86]. The chimera states can be understood as emerging naturally through a symmetry-breaking bifurcation
from the Kuramoto model's partially synchronized
state [23]. The mathematical mechanisms behind the
chimera states have been discussed by the latest efforts of [4], [87].
2.3. Explosive Synchronization
Different from the most efforts relating on the second-order transitions, a discontinuous first-order transition
(see Fig. 4), named explosive synchronization, occurs

1
Forward
Backward

0.8

(r )

0.6
0.4
0.2
0
0.8

1

1.2

1.4

1.6

1.8

(λ)
Figure 4. Order parameter r with coupling strength m [5].

θi

(a)

2.4. Cluster Synchronization
Apart from that all the oscillators eventually involve in
a single cluster, like a giant oscillator, with the common
phase, the oscillator populations may organize into
multiple groups or clusters in some special cases [101],
[102]. To be specific, each cluster consists of oscillators with the identical phase, but oscillators in different clusters have distinct phases (see Fig. 5). Focused
on the inducements for clustering, researchers have
revealed a variety of mechanisms including the higher
harmonics in the coupling function [103], noise effect
[104], adaptive couplings [105], and the Hebbian learning rate [106].
By introducing a negative coupling strength, the
phase clustering emerges when the classic Kuramoto
model (1) is modified as follows [107]:

(b)

Figure 5. Clustering of the Kuramoto oscillators. (a) Initial
state. (b) Clustering (three subsets) [107].

50 	

in scale-free Kuramoto-oscillator networks [5]. GómezGardenes et al. have claimed that the reason for this
transition is the correlation between network topology
and oscillator's dynamics [5], where the natural frequencies are positively correlated with the degrees in
the form ~ i = k i (ki is the degree of node i).
Immediately after that, Peron et al. [88] find that the
time-delayed coupling can enhance the explosive synchronization. The previous study [5] is included as a
specific case, and a general complex network is further
explored for explosive synchronization [89], where a
positive correlation between the coupling strengths of
oscillators and the absolute of their natural frequencies
is assumed. Leyva et al. [90] introduce a weighting procedure based on the link frequency mismatch and on
the link betweenness to induce an explosive transition
to synchronization. Zou et al. [91] unravel the underlying mechanisms of explosive synchronization, in which
the dynamical origin of the hysteresis is a change of the
attraction basin of synchronization state. Besides the
first-order Kuramoto models [5], [88]-[91], the nodes
in a second-order Kuramoto-oscillator network also
perform a cascade of transitions toward a synchronous
macroscopic state [92]. The latest report of [93] has
reviewed the explosive transition in exhaustive detail.
And recent researches have considered other factors
in explosive transition, such as community structure
[6], self-similarity [94], stochastic perturbations [95],
Cartesian product [96], and time-delayed coupling [97].
Furthermore, this collective phenomenon has been observed in directed [98], adaptive [99], and multilayer
[100] complex networks.

io i (t) = ~ i +

1

N

/ K ij · sin (i j (t) - i i (t)), i = 1, 2, f, N,
K max
j=1

(5)

IEEE CIRCUITS AND SYSTEMS MAGAZINE 		

THIRD QUARTER 2020
IEEE Circuits and Systems Magazine - Q3 2020

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