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
Table of Contents for the Digital Edition of IEEE Circuits and Systems Magazine - Q3 2020
Contents
IEEE Circuits and Systems Magazine - Q3 2020 - Cover1
IEEE Circuits and Systems Magazine - Q3 2020 - Cover2
IEEE Circuits and Systems Magazine - Q3 2020 - Contents
IEEE Circuits and Systems Magazine - Q3 2020 - 2
IEEE Circuits and Systems Magazine - Q3 2020 - 3
IEEE Circuits and Systems Magazine - Q3 2020 - 4
IEEE Circuits and Systems Magazine - Q3 2020 - 5
IEEE Circuits and Systems Magazine - Q3 2020 - 6
IEEE Circuits and Systems Magazine - Q3 2020 - 7
IEEE Circuits and Systems Magazine - Q3 2020 - 8
IEEE Circuits and Systems Magazine - Q3 2020 - 9
IEEE Circuits and Systems Magazine - Q3 2020 - 10
IEEE Circuits and Systems Magazine - Q3 2020 - 11
IEEE Circuits and Systems Magazine - Q3 2020 - 12
IEEE Circuits and Systems Magazine - Q3 2020 - 13
IEEE Circuits and Systems Magazine - Q3 2020 - 14
IEEE Circuits and Systems Magazine - Q3 2020 - 15
IEEE Circuits and Systems Magazine - Q3 2020 - 16
IEEE Circuits and Systems Magazine - Q3 2020 - 17
IEEE Circuits and Systems Magazine - Q3 2020 - 18
IEEE Circuits and Systems Magazine - Q3 2020 - 19
IEEE Circuits and Systems Magazine - Q3 2020 - 20
IEEE Circuits and Systems Magazine - Q3 2020 - 21
IEEE Circuits and Systems Magazine - Q3 2020 - 22
IEEE Circuits and Systems Magazine - Q3 2020 - 23
IEEE Circuits and Systems Magazine - Q3 2020 - 24
IEEE Circuits and Systems Magazine - Q3 2020 - 25
IEEE Circuits and Systems Magazine - Q3 2020 - 26
IEEE Circuits and Systems Magazine - Q3 2020 - 27
IEEE Circuits and Systems Magazine - Q3 2020 - 28
IEEE Circuits and Systems Magazine - Q3 2020 - 29
IEEE Circuits and Systems Magazine - Q3 2020 - 30
IEEE Circuits and Systems Magazine - Q3 2020 - 31
IEEE Circuits and Systems Magazine - Q3 2020 - 32
IEEE Circuits and Systems Magazine - Q3 2020 - 33
IEEE Circuits and Systems Magazine - Q3 2020 - 34
IEEE Circuits and Systems Magazine - Q3 2020 - 35
IEEE Circuits and Systems Magazine - Q3 2020 - 36
IEEE Circuits and Systems Magazine - Q3 2020 - 37
IEEE Circuits and Systems Magazine - Q3 2020 - 38
IEEE Circuits and Systems Magazine - Q3 2020 - 39
IEEE Circuits and Systems Magazine - Q3 2020 - 40
IEEE Circuits and Systems Magazine - Q3 2020 - 41
IEEE Circuits and Systems Magazine - Q3 2020 - 42
IEEE Circuits and Systems Magazine - Q3 2020 - 43
IEEE Circuits and Systems Magazine - Q3 2020 - 44
IEEE Circuits and Systems Magazine - Q3 2020 - 45
IEEE Circuits and Systems Magazine - Q3 2020 - 46
IEEE Circuits and Systems Magazine - Q3 2020 - 47
IEEE Circuits and Systems Magazine - Q3 2020 - 48
IEEE Circuits and Systems Magazine - Q3 2020 - 49
IEEE Circuits and Systems Magazine - Q3 2020 - 50
IEEE Circuits and Systems Magazine - Q3 2020 - 51
IEEE Circuits and Systems Magazine - Q3 2020 - 52
IEEE Circuits and Systems Magazine - Q3 2020 - 53
IEEE Circuits and Systems Magazine - Q3 2020 - 54
IEEE Circuits and Systems Magazine - Q3 2020 - 55
IEEE Circuits and Systems Magazine - Q3 2020 - 56
IEEE Circuits and Systems Magazine - Q3 2020 - 57
IEEE Circuits and Systems Magazine - Q3 2020 - 58
IEEE Circuits and Systems Magazine - Q3 2020 - 59
IEEE Circuits and Systems Magazine - Q3 2020 - 60
IEEE Circuits and Systems Magazine - Q3 2020 - 61
IEEE Circuits and Systems Magazine - Q3 2020 - 62
IEEE Circuits and Systems Magazine - Q3 2020 - 63
IEEE Circuits and Systems Magazine - Q3 2020 - 64
IEEE Circuits and Systems Magazine - Q3 2020 - 65
IEEE Circuits and Systems Magazine - Q3 2020 - 66
IEEE Circuits and Systems Magazine - Q3 2020 - 67
IEEE Circuits and Systems Magazine - Q3 2020 - 68
IEEE Circuits and Systems Magazine - Q3 2020 - Cover3
IEEE Circuits and Systems Magazine - Q3 2020 - Cover4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2023Q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2023Q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2023Q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2022Q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021Q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2021q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2020q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2019q1
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q4
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q3
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q2
https://www.nxtbook.com/nxtbooks/ieee/circuitsandsystems_2018q1
https://www.nxtbookmedia.com