Rössler had a brilliant and successful life as a scientist during which he published a benchmark dynamical system by using an electronic circuit interpreting chemical reactions. This is our contribution to honor his splendid erudite career. It is a hot topic to regulate a network behavior using the pinning control with respect to a small set of nodes in the network. Besides pinning to a small number of nodes, small perturbation to the node dynamics is also demanded. In this paper, the pinning synchronization of a coupled Rössler-network with time delay using univariate impulse control is investigated. Using the Lyapunov theory, a theorem is proved for the asymptotic stability of synchronization in the network. Simulation is given to validate the correctness of the analysis and the effectiveness of the proposed univariate impulse pinning controller.
REFERENCES
1.
L.
Pavel
, “Dynamics and stability in optical communication networks: A system theory framework
,” Automatica
40
(8
), 1361
–1370
(2004
). 2.
C.
Bai
, H. P.
Ren
, and G.
Kolumbán
, “Double-sub-streams M-ary differential chaos shift keying wireless communication system using chaotic shape-forming filter
,” IEEE Trans. Circuits Syst. I: Regul. Pap.
67
, 3574
–3587
(2020
). 3.
Y.
Zhang
, A. G.
Chen
, Y. J.
Tang
, J. W.
Dang
, and G. P.
Wang
, “Plaintext-related image encryption algorithm based on perceptron-like network
,” Inf. Sci.
526
, 180
–202
(2020
). 4.
H. P.
Ren
and Z.
Ma
, “License plate recognition using complex network feature,” in Proceedings of the 11th World Congress on Intelligent Control and Automation
, Shenyang, China (IEEE, 2014), pp. 5426–5431.5.
H. P.
Ren
, C.
Bai
, M. S.
Baptista
, and C.
Grebogi
, “Weak connections form an infinite number of patterns in the brain
,” Sci. Rep.
7
, 46472
(2017
). 6.
R. O.
Grigoriev
, M. C.
Cross
, and H. G.
Schuster
, “Pinning control of spatiotemporal chaos
,” Phys. Rev. Lett.
79
, 2795
–2798
(1997
). 7.
C. D.
Li
, X. F.
Liao
, and X. F.
Yang
, “Impulsive stabilization and synchronization of a class of chaotic delay systems
,” Chaos
15
, 043103
(2005
). 8.
D.
Ding
, Z.
Tang
, Y.
Wang
, and Z. C.
Ji
, “Synchronization of nonlinearly coupled complex networks: Distributed impulsive method
,” Chaos Solitons Fractals
133
, 109620
(2020
). 9.
H. P.
Ren
and C. Z.
Han
, “Parameter identification and synchronization of chaotic system using conjugate gradient method
,” Chin. J. Sci. Instrum.
29
, 792
(2008
).10.
J.
Zhou
, J. A.
Lu
, and J. H.
Lv
, “Pinning adaptive synchronization of a general complex dynamical network
,” Automatica
44
, 996
–1003
(2008
). 11.
X. S.
Yang
, J. D.
Cao
, and Z. C.
Yang
, “Synchronization of couple reaction-diffusion neural network with time-varying delays via pinning-impulsive controller
,” SIAM J. Control Optim.
51
, 3486
–3510
(2013
). 12.
W. L.
He
, F.
Qian
, and J. D.
Cao
, “Pinning-controlled synchronization of delayed neural networks with distributed-delay coupling via impulsive control
,” Neural Netw.
85
, 1
–9
(2016
). 13.
C. B.
Yi
, C.
Xu
, J. W.
Feng
, J. Y.
Wang
, and Y.
Zhao
, “Pinning synchronization for reaction-diffusion neural networks with delays by mixed impulsive control
,” Neurocomputing
339
, 270
–278
(2019
). 14.
J. Q.
Lu
, Z. D.
Wang
, J. D.
Cao
, D. W. C.
Ho
, and J.
Kurths
, “Pinning impulsive stabilization of nonlinear dynamical networks with time-varying delay
,” Int. J. Bifurcat. Chaos
22
, 1250176
(2012
). 15.
T. P.
Chen
, X. W.
Liu
, and W. L.
Lu
, “Pinning complex networks by a single controller
,” IEEE Trans. Circuits Syst.-I
54
, 1317
–1326
(2007
). 16.
K.
Tian
, C.
Bai
, H. P.
Ren
, and C.
Grebogi
, “Hyperchaos synchronization using univariate impulse control
,” Phys. Rev. E
100
, 052215
(2019
). 17.
O. E.
Rössler
, “An equation for hyperchaos
,” Phys. Lett. A
71
, 155
–157
(1979
). 18.
R.
Thomas
, V.
Basios
, M.
Eiswirth
, T.
Kruel
, and O. E.
Rössler
, “Hyperchaos of arbitrary order generated by a single feedback circuit, and the emergence of chaotic walks
,” Chaos
14
, 669
–674
(2004
). 19.
X. Q.
Wu
, Q. S.
Li
, C. Y.
Liu
, J.
Liu
, and C. W.
Xie
, “Synchronization in duplex network of coupled Rössler oscillators with different inner-coupling matrices,” Neurocomputing
408
, 31
–41
(2020
).20.
I.
Bodale
and V. A.
Oancea
, “Chaos control for Willamowski-Rössler model of chemical reactions
,” Chaos Solitons Fractals
78
, 1
–9
(2015
). 21.
Q. D.
Li
, “A topological horseshoe in the hyperchaotic Rössler attractor
,” Phys. Lett. A
372
, 2989
–2994
(2008
). 22.
K.
Tian
, H. P.
Ren
, and C.
Bai
, “Synchronization of hyperchaos with time delay using impulse control
,” IEEE Access
8
, 72570
–72576
(2020
). 23.
H. W.
Kuhn
, “The Hungarian method for the assignment problem
,” Nav. Res. Logist. Q.
2
, 83
–97
(1955
). 24.
J.
Munkres
, “Algorithms for assignment and transportation problems
,” J. Soc. Ind. Appl. Math.
5
, 32
–38
(1957
). 25.
K.
Pyragas
, “Continuous control of chaos by self-controlling feedback
,” Phys. Lett. A
170
, 421
–428
(1992
). 26.
H. P.
Ren
, D.
Liu
, and C. Z.
Han
, “Anticontrol of chaos via direct time delay feedback
,” Acta Phys. Sin.
6
, 2694
(2006
).27.
X. Z.
Liu
and G.
Ballinger
, “Uniform asymptotic stability of impulsive delay differential equation
,” Comput. Math. Appl.
41
, 903
–915
(2001
). 28.
L. A.
Aguirre
and C.
Letellier
, “Controllability and synchronizability: Are they related?
,” Chaos Solitons Fractals
83
, 242
–251
(2016
). 29.
Y. Y.
Liu
, J. J.
Slotine
, and A. L.
Barabasi
, “Controllability of complex networks
,” Nature
473
, 167
–173
(2011
). 30.
C. T.
Lin
, “Structural controllability
,” IEEE Trans. Automat. Control
19
, 201
–208
(1974
). 31.
C.
Letellier
, I.
Sendiña-Nadal
, and L. A.
Aguirre
, “A nonlinear graph-based theory for dynamical network observability
,” Phys. Rev. E
98
, 020303
(2018
). 32.
R.
Hermann
and A. J.
Krener
, “Nonlinear controllability and observability
,” IEEE Trans. Automat. Control
22
, 728
–740
(1977
). 33.
H. P.
Ren
, C.
Bai
, K.
Tian
, and C.
Grebogi
, “Dynamics of delay induced composite multi-scroll attractor and its application in encryption
,” Int. J. Non Linear Mech.
94
, 334
–342
(2017
). 34.
J. D.
Farmer
, “Chaotic attractors of an infinite-dimensional dynamical system
,” Physica D
4
, 366
–393
(1982
). 35.
P.
Grassberger
and I.
Procaccia
, “Measuring the strangeness of strange attractors
,” Physica D
9
, 189
–208
(1983
). © 2020 Author(s).
2020
Author(s)
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