IEEE Circuits and Systems Magazine - Q3 2020 - 57

/ R (i )

i=1

N

, (24)

where R(i) is the local order parameter of node i over
the ensemble of its neighbor(s) in one layer and its counterpart in the other layer. R global has the same meaning
as defined in (2).
Via simulating in a duplex Li network [155], Figure 12(a)
shows that the inter-layer coupling strength is a very
important factor influencing the inter-layer synchronization [156]. For the small intra-layer coupling strength,
the red area at the lower right corner of Fig. 12(a) implies
that the inter-layer synchronization emerges -regardless
of the global synchronization [156].
Furthermore, the global complete synchronization
is achieved if the intra-layer synchronization and interlayer synchronization occur together. The conventional
global order parameter R global can be used to measure
the degree of global complete synchronization, and the
simulation results are illustrated in Fig. 12(b). If the intra-layer or inter-layer coupling strength is very small,
the duplex Kuramoto-oscillator network (23) cannot
achieve global complete synchronization [156]. That is,
the global complete synchronization appears only when
both the intra-layer and inter-layer coupling strengths
exceed their threshold values.

-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

2.5

(λintra)

2
1.5
1
0.5
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
(λinter)
(a)

lim
(i i - i j) = 0, 6i, j = 1, 2, f, N.
t"3
Frequency synchronization is achieved for nonidentical Kuramoto-oscillator case, i.e., ~ i ! ~ j (i ! j ), if
lim
(io i - io j) = 0, 6i, j = 1, 2, f, N.
t"3
In terms of the prototypical model (1), Chopra and
Spong [45] for the first time probe into the exponential frequency synchronization in the finite oscillator
Kuramoto model with different natural frequencies,
where a lower bound (than previous work [159]) on
the coupling gain is derived. But the phase differences,
|i i - i j| , 6i, j = 1, 2, f, N, are restricted within r/2. After
that, Ha et al. [46] discuss the asymptotic complete
phase-frequency synchronization with presenting sufficient conditions. For identical oscillators, the complete phase agreement will occur exponentially for the
initial phase configurations located on the half circle.
For the non-identical oscillators, the complete frequency synchronization will occur exponentially fast,
in which the initial phase configurations still must
be less than r/2. Then, the asymptotic formation and

10

1
0.9

8
Rinter - Rglobal

3

5.1. Complete Synchronization
In a Kuramoto-oscillator network, the complete synchronization covers phase agreement and frequency synchronization, where two definitions are given as follows:
Phase agreement is achieved for identical Kuramoto-oscillator case, i.e., ~ i = ~ j, if

0.8
0.7

(λintra)

R inter =

	

5. Stability Analysis of Kuramoto-Oscillator
Networks

6

0.6
0.5

4

0.4

Rglobal

strength at layer 1 (2), k i, 1 ^k i, 2 h is the degree of node i,
a n d m 12 = m 21 = m inter i s t h e i n t e r - l a y e r c o u p l i n g
strength. K i represents the set of neighbors of node i. In
order to measure the level of inter-layer synchronization,
the inter-layer order parameter R inter is put forward
as follows

0.3
0.2

2

0.1
0

0
0

1

2
3
(λinter)

4

5

(b)

Figure 12. (a) R inter - R global varying as a function of inter-layer coupling strength m inter and intra-layer coupling strength m intra . (b)
The color variation shows the value of global order parameter R global as a function of inter-layer coupling strength m inter and intralayer coupling strength m intra [156].

THIRD QUARTER 2020 		

IEEE CIRCUITS AND SYSTEMS MAGAZINE	

57



IEEE Circuits and Systems Magazine - Q3 2020

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