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Research Papers

Hidden Chaotic Attractors and Synchronization for a New Fractional-Order Chaotic System

[+] Author and Article Information
Zuoxun Wang

Department of Electrical Engineering
and Automation,
Shandong Academy of Sciences,
Qilu University of Technology,
No. 3501 Daxue Road,
Jinan 250353, China
e-mail: wangzuoxun@126.com

Jiaxun Liu

Department of Electrical Engineering
and Automation,
Shandong Academy of Sciences,
Qilu University of Technology,
No. 3501 Daxue Road,
Jinan 250353, China
e-mail: liujiaxunliujiaxun@163.com

Fangfang Zhang

Department of Electrical Engineering
and Automation,
Shandong Academy of Sciences,
Qilu University of Technology,
No. 3501 Daxue Road,
Jinan 250353, China
e-mail: zhff4u@163.com

Sen Leng

Department of Electrical Engineering
and Automation,
Shandong Academy of Sciences,
Qilu University of Technology,
No. 3501 Daxue Road,
Jinan 250353, China
e-mail: ls119@foxmail.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 22, 2018; final manuscript received April 23, 2019; published online June 10, 2019. Assoc. Editor: Firdaus Udwadia.

J. Comput. Nonlinear Dynam 14(8), 081010 (Jun 10, 2019) (11 pages) Paper No: CND-18-1568; doi: 10.1115/1.4043670 History: Received December 22, 2018; Revised April 23, 2019

Although a large number of hidden chaotic attractors have been studied in recent years, most studies only refer to integer-order chaotic systems and neglect the relationships among chaotic attractors. In this paper, we first extend LE1 of sprott from integer-order chaotic systems to fractional-order chaotic systems, and we add two constant controllers which could produce a novel fractional-order chaotic system with hidden chaotic attractors. Second, we discuss its complicated dynamic characteristics with the help of projection pictures and bifurcation diagrams. The new fractional-order chaotic system can exhibit self-excited attractor and three different types of hidden attractors. Moreover, based on fractional-order finite time stability theory, we design finite time synchronization scheme of this new system. And combination synchronization of three fractional-order chaotic systems with hidden chaotic attractors is also derived. Finally, numerical simulations demonstrate the effectiveness of the proposed synchronization methods.

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Figures

Grahic Jump Location
Fig. 1

Bifurcation diagram of the new fractional-order chaotic systems with m =0, n =0, x(0) = 0.1, y(0) = 0.5, and z(0) = 0.5

Grahic Jump Location
Fig. 2

Strange attractor of the new fractional-order chaotic system with m =0, n =0, q =0.992, x(0) = 0.1, y(0) = 0.5, and z(0) = 0.5: (a) projection in xz plane, (b) projection in xy plane, (c) projection in yz plane, and (d) 3D attractors

Grahic Jump Location
Fig. 3

Bifurcation diagram of the new fractional-order chaotic system with m =0.004, n =0, x(0) = 0.1, y(0) = 0.5, and z(0) = 0.5

Grahic Jump Location
Fig. 4

Strange attractor of the new fractional-order chaotic system with m =0.004, n =0, and q =0.991: (a) projection in xz plane, (b) projection in xy plane, (c) projection in yz plane, and (d) 3D attractors

Grahic Jump Location
Fig. 5

Bifurcation diagram of the new fractional-order chaotic systems with m = –0.004, n =0, x(0) = 0.1, y(0) = 0.5, and z(0) = 0.5

Grahic Jump Location
Fig. 6

Strange attractor of the new fractional-order chaotic system for m = –0.004, n =0, q =0.993: (a) projection in xz plane, (b) projection in xy plane, (c) projection in yz plane, and (d) 3D attractors

Grahic Jump Location
Fig. 7

Bifurcation diagram of the new fractional-order chaotic system with m =0, n =0.004, x(0) = 0.1, y(0) = 0.5, and z(0) = 0.5

Grahic Jump Location
Fig. 8

Strange attractor of the new fractional-order chaotic system with m =0, n =0.004, and q =0.9905: (a) projection in xz plane, (b) projection in xy plane, (c) projection in yz plane, and (d) 3D attractors

Grahic Jump Location
Fig. 9

The error curves for finite time synchronization with q =0.99, k1 = 9, and B1 = 0.8: (a) error e1 = x1x, (b) error e2 = y1y, and (c) error e3 = z1z

Grahic Jump Location
Fig. 10

The phase plots of state variables for finite time synchronization with controller (17): (a) state variables x1 and x, (b) state variables y1 and y, and (c) state variables z1 and z

Grahic Jump Location
Fig. 11

The plots of the logarithm of the absolute value of the errors as a function of time with controller (17): (a) the plot of the logarithm of the absolute value of e1, (b) the plot of the logarithm of the absolute value of e2, and (c) the plot of the logarithm of the absolute value of e3

Grahic Jump Location
Fig. 12

The error curves for combination synchronization with q =0.99, k2 = 12, and B2 = 0.8: (a) error curve of e1 = x3x2x1, (b) error curve of e2 = y3y2y1, and (c) error curve of e3 = z3z2z1

Grahic Jump Location
Fig. 13

The phase plots of state variables for combination synchronization with controller (29), (a) state variables x1 + x2 and x3, (b) state variables y1 + y2 and y3, and (c) state variables z1 + z2 and z3

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