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Research Papers

Synchronization for Incommensurate Riemann–Liouville Fractional-Order Time-Delayed Competitive Neural Networks With Different Time Scales and Known or Unknown Parameters1

[+] Author and Article Information
Yajuan Gu

Department of Mathematics,
Beijing Jiaotong University,
Beijing 100044, China
e-mail: 16118413@bjtu.edu.cn

Hu Wang

School of Statistics and Mathematics,
Central University of Finance and Economics,
Beijing 100081, China
e-mail: wanghu1985712@163.com

Yongguang Yu

Department of Mathematics,
Beijing Jiaotong University,
Beijing 100044, China
e-mail: ygyu@bjtu.edu.cn

2Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 7, 2018; final manuscript received December 26, 2018; published online February 15, 2019. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 14(5), 051002 (Feb 15, 2019) (11 pages) Paper No: CND-18-1395; doi: 10.1115/1.4042494 History: Received September 07, 2018; Revised December 26, 2018

Synchronization for incommensurate Riemann–Liouville fractional competitive neural networks (CNN) with different time scales is investigated in this paper. Time delays and unknown parameters are concerned in the model, which is more practical. Two simple and effective controllers are proposed, respectively, such that synchronization between the salve system and the master system with known or unknown parameters can be achieved. The methods are more general and less conservative which can also be applied to commensurate integer-order systems and commensurate fractional systems. Furthermore, two numerical ensamples are provided to show the feasibility of the approach. Based on the chaotic masking method, the example of chaos synchronization application for secure communication is provided.

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Figures

Grahic Jump Location
Fig. 1

Chaotic behavior of system (28): (a) X1(t)−X2(t) and (b) S1(t)−S2(t)

Grahic Jump Location
Fig. 2

Trajectories of master system (28) and salve system (29): (a) trajectories of X1(t) and Y1(t), (b) trajectories of X2(t)  and  Y2(t), (c) trajectories of S1(t)  and  R1(t), and (d) trajectories of S2(t)  and  R2(t)

Grahic Jump Location
Fig. 3

Synchronization errors of master system (28) and salve system (29): (a) synchronization errors of Xi(t)  and  Yi(t) and (b) synchronization errors of Si(t)  and  Ri(t)

Grahic Jump Location
Fig. 4

Trajectories of master system (28) and salve system (30): (a) trajectories of X1(t) and Y1(t), (b) trajectories of X2(t)  and  Y2(t), (c) trajectories of S1(t)  and  R1(t), and (d) trajectories of S2(t)  and  R2(t)

Grahic Jump Location
Fig. 5

Synchronization errors of master system (28) and salve system (30): (a) synchronization errors of Xi(t)  and  Yi(t) and (b) synchronization errors of Si(t)  and  Ri(t)

Grahic Jump Location
Fig. 6

Time evolution of parameters and controlling strengths: (a) time evolution of parameters ai(t),Bi(t)  and  di(t), (b) time evolution of parameters bij(t), (c) time evolution of parameters cij(t), and (d) time evolution of controlling strengths μi(t)

Grahic Jump Location
Fig. 7

Diagram of secure communication in virtue of chaos synchronization

Grahic Jump Location
Fig. 8

Secure communication in virtue of chaos synchronization with chaos masking strategy: (a) state trajectory of information signal I(t), (b) state trajectory of transmitted signal T(t), (c) state trajectory of recovered signal R(t), and (d) error between recovered signal R(t) and information signal I(t)

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