IEEE Circuits and Systems Magazine - Q3 2020 - 63

Future researches of Kuramoto-oscillator networks may cover more factors including,
to name a few but not limited to, multiplex connectivity, heterogeneous couplings,
and time-evolution dynamics with both the practical support of stability
theory and potential applications in engineering and sociology.
on multi-layer networks contained the Kuramoto
oscillators. Future directions can turn to investigate the entanglement mechanism between
multi-layer networks, where distinct layers can
possess different dynamical systems, and the
relationship between intra-layer and inter-layer
nodes may be cooperative, competitive, decoupled, or others. Furthermore, the major
results so far obtained on multi-layer Kuramoto-oscillator networks are qualitative, which is
based on the statistical mechanics and a large
number of numerical simulations. Quantitative
and analytical results of multi-layer Kuramotooscillator networks deserve more attention and
efforts in the literature.
■■ Through rigorous stability analysis on Kuramotooscillator networks, past efforts have derived various sufficient conditions to guarantee the desired
synchronization. In the continued works, researchers can devote to relax these stringent conditions
and consider more cases of network topology so as
to cater to broad applications. For the detrimental
case that the synchronization of Kuramoto-oscillator networks can not be achieved spontaneously
due to the sparse links, dispersed initial phases,
and weak coupling strength, it is necessary to design novel controllers to guarantee the synchronization. Meanwhile, in the process of realizing the
finite-time, fixed-time, and global stochastic synchronization of Kuramoto-oscillator networks, the
future steps include to reduce the number of control layers and/or controlled nodes for saving the
control cost, such as adopting the pinning control
[178], [72], or introducing a pacemaker.
Acknowledgment
This work was partly supported by the National Natural Science Foundation of China (No. 71731004, No.
61751303) and the Natural Science Fund of Distinguished
Young Scholarship of China (No. 61425019).
Jie Wu (S'19) received the B.S. degree in
applied mathematics from Hainan
Normal University, Haikou, the M.S. degree in mathematics from China University of Mining and Technology, Xuzhou,
and the Ph.D. degree in circuits and sysTHIRD QUARTER 2020

tems from Fudan University, Shanghai, China, in 2013,
2015, 2020, respectively.
He is a reviewer of SIAM Journal on Control and
Optimization. His current research interests include
sy nchronization of complex networks and control
of networks.
Xiang Li (M'05-SM'08) received the BS
and PhD degrees in control theory and
control engineering from Nankai University, China, in 1997 and 2002, respectively. Before joining Fudan University
as a professor of Electronic Engineering
Department in 2008, he was with City University of Hong
Kong, Int. University Bremen and Shanghai Jiao Tong
University as post-doc research fellow, Humboldt research fellow and an associate professor in 2002-2004,
2005-2006 and 2004-2007, respectively. He served as
head of the Electronic Engineering Department at Fudan
University in 2010-2015. Currently, he is a distinguished
professor of Fudan University, and chairs the Adaptive
Networks and Control (CAN) group and the Research
Center of Smart Networks & Systems, School of Information Science & Engineering, Fudan University. He served
as several associate editor including the IEEE Transactions on Circuits and Systems-I: Regular Papers (2010-
2015), and serves as the associate editor of IEEE Transactions on Network Science and Engineering, Journal of
Complex Networks, and IEEE Circuits and Systems Society Newsletter. His main research interests cover network science and system control in both theory and applications. He has (co-)authored 4 research monographs,
6 academic chapters, and more than 150 peer-refereed
publications in journals and conferences. He received the
IEEE Guillemin-Cauer Best Transactions Paper Award
from the IEEE Circuits and Systems Society in 2005, Shanghai Natural Science Award (1st class) in 2008, Shanghai
Science and Technology Young Talents Award in 2010, National Science Foundation for Distinguished Young Scholar of China in 2014, National Natural Science Award (2nd
class) in 2015, Ten-Thousand Talent Program of China in
2017, TCCT CHEN Han-Fu Award of Chinese Automation
Association in 2019, among other awards and honors.
References
[1] Y. Kuramoto, "Self-entrainment of a population of coupled non-linear oscillators," in Proc. Int. Symp. Mathematical Problems Theoretical
Physics. Berlin: Springer-Verlag, 1975.
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