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Research Papers

A Novel Adaptive Active Control Projective Synchronization of Chaotic Systems

[+] Author and Article Information
Boan Quan

School of Information Science and Engineering,
Hunan University,
Changsha 410082, China
e-mail: quanboan@163.com

Chunhua Wang

School of Information Science and Engineering,
Hunan University,
Changsha 410082, China
e-mail: wch1227164@hnu.edu.cn

Jingru Sun

School of Information Science and Engineering,
Hunan University,
Changsha 410082, China
e-mail: jt_sunjr@hnu.edu.cn

Yilin Zhao

School of Information Science and Engineering,
Hunan University,
Changsha 410082, China
e-mail: zhaoyl1992@126.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received May 17, 2017; final manuscript received January 20, 2018; published online March 23, 2018. Assoc. Editor: Bogdan I. Epureanu.

J. Comput. Nonlinear Dynam 13(5), 051001 (Mar 23, 2018) (9 pages) Paper No: CND-17-1219; doi: 10.1115/1.4039189 History: Received May 17, 2017; Revised January 20, 2018

This paper investigates adaptive active control projective synchronization scheme. A general synchronization controller and parameter update laws are proposed to stabilize the error system for the identical structural chaotic systems. It is the first time that the active synchronization, the projective synchronization, and the adaptive synchronization are combined to achieve the synchronization of chaotic systems, which extend the control capability of achieving chaotic synchronization. By using a constant diagonal matrix, the active control is developed. Especially, when designing the controller, we just need to ensure that the diagonal elements of the diagonal matrix are less than or equal 0. So, the synchronization of chaotic systems can be realized more easily. Furthermore, by proposing an active controller, in combination with several different control schemes, we lower the complexity of the design process of the controller. More importantly, the larger the absolute value of product of the diagonal elements of diagonal matrix is, the smoother the curve of chaotic synchronization is and the shorter the time of chaotic synchronization is. In our paper, we take Lorenz system as an example to verify the effectiveness of the proposed synchronization scheme. Theoretical analysis and numerical simulations demonstrate the feasibility of this control method.

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References

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Figures

Grahic Jump Location
Fig. 1

The phase diagram of system (21): (a) the projection of the attractor of x–y plane, and (b) the projection of the attractor of x-z plane, and (c) the projection of the attractor of y–z plane, and (d) The projection of the attractor of x-y-z plane

Grahic Jump Location
Fig. 2

The simulations of the synchronization between the drive system (22) and the response system (23) when C = diag{−1, −2, −3}: (a) the synchronization between y1tand2x1(t), and (b) the synchronization between y2tand2x2(t), and (c) the synchronization between y3tand2x3(t)

Grahic Jump Location
Fig. 3

The simulations of the synchronization between the drive system (22) and the response system (23) when C = diag{−2, −2, −2}: (a) the synchronization between y1tand2x1(t), and (b) the synchronization between y2tand2x2(t), and (c) the synchronization between y3tand2x3(t)

Grahic Jump Location
Fig. 4

The simulations of the synchronization between the drive system (22) and the response system (23) when C = diag{−3, −3, −3}: (a) the synchronization between y1tand2x1t, and (b) the synchronization between y2tand2x2t, and (c) the synchronization between y3tand2x3t

Grahic Jump Location
Fig. 5

The simulations of the errors between the drive system (22) and the response system (23) when C = diag{−1, −2, −3}: (a) the evolution of e1, (b) the evolution of e2, and (c) the evolution of e3

Grahic Jump Location
Fig. 6

The simulations of the errors between the drive system (22) and the response system (23) when C = diag{−2, −2, −2}: (a) the evolution of e1, (b) the evolution of e2, and (c) the evolution of e3

Grahic Jump Location
Fig. 7

The simulations of the errors between the drive system (22) and the response system (23) when C = diag{−3, −3, −3}: (a) the evolution of e1, (b) the evolution of e2, and (c) the evolution of e3

Grahic Jump Location
Fig. 8

The identifications of a1t,b1t,c1t when C = diag{−1, −2, −3}: (a) the identification of a1t, (b) the identification of b1t, and (c) the identification ofc1t

Grahic Jump Location
Fig. 9

The identifications of a1t,b1t,c1(t) when C = diag{−2, −2, −2}: (a) the identification of a1(t), (b) the identification of b1(t), and (c) the identification ofc1(t)

Grahic Jump Location
Fig. 10

The identifications of a1t,b1t,c1(t) when C = diag{−3, −3, −3}: (a) the identification of a1(t), (b) the identification of b1(t), and (c) the identification ofc1(t)

Grahic Jump Location
Fig. 11

The simulations of the errors between the drive system (22) and the response system (23) when C = diag{−1, −2, −3}, and the scaling factor α=2: (a) the evolution of, e1, (b) the evolution of e2, and (c) the evolution of e3

Grahic Jump Location
Fig. 12

The simulations of the errors between the drive system (22) and the response system (23) when C = diag{−1, −2, −3}, and the scaling factor α=3: (a) the evolution of, e1, (b) the evolution of, e2, and (c) the evolution of e3

Grahic Jump Location
Fig. 13

The simulations of the errors between the drive system (22) and the response system (23) when C = diag{−1, −2, −3}, and the scaling factor α=4: (a) the evolution of e1, (b) the evolution of, e2, and (c) the evolution of e3

Grahic Jump Location
Fig. 14

The simulations of the errors between the drive system (22) and the response system (23) when C = diag{−1, −1, −1} and α=2: (a) the evolution of, e1, (b) the evolution of, e2, and (c) the evolution of e3

Grahic Jump Location
Fig. 15

The simulations of the errors between the drive system (22) and the response system (23) when C = diag{−4, −4, −4} and α=2: (a) the evolution of, e1, (b) the evolution of e2, and (c) the evolution of e3

Grahic Jump Location
Fig. 16

The simulations of the errors between the drive system (22) and the response system (23) when C=diag{−25, −25, −25} and α=2: (a) the evolution of e1, (b) the evolution of e2, and (c) the evolution of e3

Grahic Jump Location
Fig. 17

The simulations of the errors between the drive system (22) and the response system (23) when C = diag{−200, −200, −200} and α=10: (a) the evolution of e1, (b) the evolution of e2, (c) the evolution of e3, and (d) the simulations of the errors [28]

Grahic Jump Location
Fig. 18

The simulations of the errors between the drive system (22) and the response system (23) when C = diag{−200, −200, −200} and α=10: (a) the evolution of e1, (b) the evolution ofe2, (c) the evolution of e3, and (d) the simulations of the errors[29]

Grahic Jump Location
Fig. 19

The simulations of the errors between the drive system (22) and the response system (23) when C = diag{−200, −200, −200} and α=10: (a) the evolution of e1, (b) the evolution ofe2, (c) the evolution of e3, and (d) The simulations of the errors[30]

Grahic Jump Location
Fig. 20

The simulations of the errors between the drive system (22) and the response system (23) when C = diag{−200, −200, −200} and α=10: (a) the evolution of e1, (b) the evolution ofe2, (c) the evolution of e3, and (d) the simulations of the errors[31]

Grahic Jump Location
Fig. 21

The simulations of the errors between the drive system (22) and the response system (23) when C = diag{−200, −200, −200} and α=10: (a) the evolution of e1, (b) the evolution of e2, (c) the evolution of e3, and (d) the simulations of the errors [32]

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