IEEE Circuits and Systems Magazine - Q3 2020 - 49

	

Kc =

2

rg (~r )

1
(r )
0

Kc

K

Figure 2. Dependence of the coherence r on coupling
strength K [30].

π

Phase

very weak, then each oscillator rotates with its intrinsic frequency ~ i, where the magnitude r of the
order parameter is always near zero. It is obvious
that this state is stable.
■■ Partially-synchronized state. As the coupling strength
K increases to the critical value Kc (also named
the critical coupling), the previous stable incoherent state is broken. Then, with the continuous
increase of coupling strength, i.e., K 2 K c , a synchronized cluster emerges with more and more
oscillators approaching, in which the magnitude
r(t) grows exponentially.
■■ Synchronized state. When the coupling strength
is sufficiently large, i.e., K & K c , all the oscillators
will arrive at a common phase, namely, phase synchronization has been achieved, where the synchronized cluster looks like a giant oscillator and
the magnitude r (t) . 1. Obviously, this synchronized state is stable, and all the oscillators share
the same frequency.
It is easy to find that the threshold (or critical coupling) Kc plays a vital role in the phase transition of all
the oscillators from incoherence to synchronization. In
Kuramoto's earliest work [2], [71], the threshold
, (4)

is obtained by analyzing model (1), if the following two
conditions hold simultaneously: (i) the probability density g (~) is unimodal and symmetric centered at its
mean frequency ~ = ~r ; (ii) the number of oscillators
tends to infinity N " 3.
In more complicated and practical situations with
network complexity addressed, numerous necessary,
-sufficient, implicit, and explicit estimations of the critical coupling strength Kc have been derived for both the
onset as well as the ultimate stage of synchronization
[35]-[37], [45]-[48]. Verwoerd and Mason [72] have
extended the results on the existence of global phaselocked states on a complete graph to the case of a complete bipartite graph, where the value of the critical coupling coefficient can be determined by solving a system
of two nonlinear equations. After that, Dörfler and Bullo
[35] provide an explicit necessary and sufficient condition on the critical coupling strength to achieve the
exponential synchronization in the finite-dimensional
Kuramoto model for an arbitrary distribution of the intrinsic frequencies ~ i. Specifically, collective synchronization occurs for K 2 K c = ~ max - ~ min , where ~ max and
~ min are the maximum and minimum intrinsic frequencies [35], respectively.
Nevertheless, the constant critical coupling strength
to achieve synchronization has an inherent defect that it
THIRD QUARTER 2020 		

-π

0

x (Space)

1

Figure 3. Chimera states in a one-dimensional periodic
space [81].

is difficult to be set appropriately in advance. To be specific, if the coupling strength is too large, it will cause
unnecessary cost, whereas it is too small to achieve
the desired synchronization. The latest efforts of [36],
[37] have overcome this drawback by adopting adaptive
couplings, in which each coupling term is designed as a
function of phase differences.
2.2. Chimera States
In the literature of Kuramoto-oscillator networks [3], [4],
[73], chimera states refer to an array of identical oscillators splitting into coexisting regions of coherent phase
agreement and the incoherent desynchronization (see
Fig. 3). Martens [74] has pointed out that the existence
of chimera states depends strongly on the underlying
network structures. The chimera states will appear via
designing non-locally [75], delay [76], and frequencyweighed [77] couplings, as well as the oscillators distributed along an infinite plane [78], a torus [79], a sphere
[80], which has been discussed in [81].
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