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Research Papers

Small-World Outer Synchronization of Small-World Chaotic Networks

[+] Author and Article Information
A. Arellano-Delgado

CONACYT-UABC,
Baja California Autonomous University,
Ensenada 22860, Baja California, Mexico
e-mail: adrian.arellano@uabc.edu.mx

R. M. López-Gutiérrez

Engineering, Architecture, and Design Faculty,
Baja California Autonomous University,
Ensenada 22860, Baja California, Mexico
e-mail: roslopez@uabc.edu.mx

R. Martínez-Clark

Electronics and Telecommunications Department,
Scientific Research and Advanced Studies
Center of Ensenada,
Ensenada 22860, Baja California, Mexico
e-mail: rigomar@cicese.edu.mx

C. Cruz-Hernández

Electronics and Telecommunications Department,
Scientific Research and Advanced Studies
Center of Ensenada,
Ensenada 22860, Baja California, Mexico
e-mail: ccruz@cicese.mx

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 12, 2018; final manuscript received July 20, 2018; published online August 22, 2018. Assoc. Editor: Firdaus Udwadia.

J. Comput. Nonlinear Dynam 13(10), 101008 (Aug 22, 2018) (11 pages) Paper No: CND-18-1159; doi: 10.1115/1.4041032 History: Received April 12, 2018; Revised July 20, 2018

In this work, small-world outer synchronization of coupled small-world networks is presented. In particular, we use Newman and Watts model to achieve small-world outer synchronization of small-world chaotic networks with Chua's oscillators like chaotic nodes. By means of extensive numerical simulations, we show that the new outer connections between existing networks decrease the necessary coupling strength to achieve outer synchronization. Two scenarios of interest are studied, (i) small-world outer synchronization with unidirectional outer connections (with chaotic master network), and (ii) small-world outer synchronization with bidirectional outer connections (without chaotic master network). In both scenarios, the isolated networks are bidirectionally coupled using Chua's oscillators like chaotic nodes.

Copyright © 2018 by ASME
Topics: Synchronization
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Figures

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Fig. 1

Two outwardly unidirectionally coupled small-world networks: (a) Np = 1 and (b) Np = 3

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Fig. 2

Synchronization diagram for outwardly unidirectionally coupled small-world networks Np versus c2

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Fig. 3

Chaotic dynamics between small-world networks A and B with master chaotic network and with Np = 1: xi1(t) (top), xi2(t) (middle), and xi3(t) (bottom)

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Fig. 4

Outer synchronizations errors of small-world networks A and B with master chaotic network A and with Np = 1: x11(t) − xi1(t) (top), x12(t) − xi2(t) (middle), and x13(t) − xi3(t) (bottom)

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Fig. 5

Phase portraits of small-world networks A and B with master chaotic network A and with Np = 1: x11 versus x1iaverage (left), x12 versus xi2 average (middle), and x13 versus xi3 average (right)

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Fig. 6

Outer synchronization errors of small-world networks A and B with master chaotic network A and Np = 3: x11(t) − xi1(t) (top), x12(t) − xi2(t) (middle), and x13(t) − xi3(t) (bottom)

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Fig. 7

Two outwardly bidirectionally coupled small-world networks: (a) Np = 1 and (b) Np = 3

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Fig. 8

Synchronization diagram for outwardly bidirectionally coupled small-world networks Np versus c2

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Fig. 9

Chaotic dynamics between small-world networks A and B without master chaotic network and with Np = 1: xi1(t) (top), xi2(t) (middle), and xi3(t) (bottom)

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Fig. 10

Outer synchronizations errors of small-world networks A and B without master chaotic network and with Np = 1: x11(t) − xi1(t) (top), x12(t) − xi2(t) (middle), and x13(t) − xi3(t) (bottom)

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Fig. 11

Phase portraits of small-world networks A and B without master chaotic network and with Np = 1: x11 versus x1iaverage (left), x12 versus xi2 average (middle), and x13 versus xi3 average (right)

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Fig. 12

Outer synchronizations errors of small-world networks A and B without master chaotic network and Np = 3: (a) x11(t) − xi1(t) (top), x12(t) − xi2(t) (middle), and x13(t) − xi3(t) (bottom)

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Fig. 13

State trajectories xi for networks A and B unidirectionally outwardly coupled with Np = 1, for t < 50, c1 = 0 and c2 = 0, for 50 ≤ t < 100, c1 = 50, and c2 = 0, and for 100 ≤ t ≤ 200, c1 = 50, and c2 = 45)

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Fig. 14

State trajectories xi for networks A and B unidirectionally outwardly coupled with Np = 3, for t < 50, c1 = 0 and c2 = 0, for 50 ≤ t < 100, c1 = 50 and c2 = 0, and for 100 ≤ t ≤ 200, c1 = 50 and c2 = 45)

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Fig. 15

State trajectories xi for networks A and B bidirectionally outwardly coupled with Np = 1, for t < 50, c1 = 0 and c2 = 0, for 50 ≤ t < 100, c1 = 50 and c2 = 0, and for 100 ≤ t ≤ 200, c1 = 50 and c2 = 45)

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Fig. 16

State trajectories xi for networks A and B bidirectionally outwardly coupled with Np = 3, for t < 50, c1 = 0 and c2 = 0, for 50 ≤ t < 100, c1 = 50 and c2 = 0, and for 100 ≤ t ≤ 200, c1 = 50 and c2 = 45)

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Fig. 17

Two outwardly coupled small-world networks with Np = 10: (a) unidirectionally and (b) bidirectionally

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Fig. 18

Synchronization diagram for two outwardly unidirectionally coupled small-world networks, Np versus c2 with N = 50

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Fig. 19

Synchronization diagram for two outwardly bidirectionally coupled small-world networks, Np versus c2 with N = 50

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Fig. 20

Chaotic dynamics between small-world networks A and B with master chaotic network and with N = 50 and Np = 10: xi1(t) (top), xi2(t) (middle), and xi3(t) (bottom)

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Fig. 21

Outer synchronizations errors of small-world networks A and B with master chaotic network with N = 50 and Np = 3: x11(t) − xi1(t) (top), x12(t) − xi2(t) (middle), and x13(t) − xi3(t) (bottom)

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Fig. 22

Chaotic dynamics between small-world networks A and B without master chaotic network and with N = 50 and Np = 10: xi1(t) (top), xi2(t) (middle), and xi3(t) (bottom)

Grahic Jump Location
Fig. 23

Outer synchronizations errors of small-world networks A and B without master chaotic network with N = 50 and Np = 3: x11(t) − xi1(t) (top), x12(t) − xi2(t) (middle), and x13(t) − xi3(t) (bottom)

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