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Research Papers

A Time-Delayed Hyperchaotic System Composed of Multiscroll Attractors With Multiple Positive Lyapunov Exponents

[+] Author and Article Information
Yue Wang

College of Computer Science and
Electronic Engineering,
Hunan University,
Changsha 410082, China
e-mail: yuewang@hnu.edu.cn

Chunhua Wang

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

Ling Zhou

College of Computer Science and
Electronic Engineering,
Hunan University,
Changsha 410082, China
e-mail: zhouling0340@163.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February 27, 2017; final manuscript received May 10, 2017; published online July 12, 2017. Assoc. Editor: Zaihua Wang.

J. Comput. Nonlinear Dynam 12(5), 051029 (Jul 12, 2017) (8 pages) Paper No: CND-17-1092; doi: 10.1115/1.4036831 History: Received February 27, 2017; Revised May 10, 2017

The paper proposes a time-delayed hyperchaotic system composed of multiscroll attractors with multiple positive Lyapunov exponents (LEs), which are described by a three-order nonlinear retarded type delay differential equation (DDE). The dynamical characteristics of the time-delayed system are far more complicated than those of the original system without time delay. The three-order time-delayed system not only generates hyperchaotic attractors with multiscroll but also has multiple positive LEs. We observe that the number of positive LEs increases with increasing time delay. Through numerical simulations, the time-delayed system exhibits a larger number of scrolls than the original system without time delay. Moreover, different numbers of scrolls with variable delay and coexistence of multiple attractors with a variable number of scrolls are also observed in the time-delayed system. Finally, we setup electronic circuit of the proposed system, and make Pspice simulations to it. The Pspice simulation results agree well with the numerical results.

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References

Figures

Grahic Jump Location
Fig. 1

LEs of the nontime-delayed system (3) and time-delayed system (4) when initial conditions (x(0), y(0), z(0)) = (1, 1, 1): (a) LEs of the system (4), (b) LEs of the system (4), and (c) LEs of the system (3)

Grahic Jump Location
Fig. 2

Hyperchaotic attractors with multiscroll of the time-delayed system (4) when t = 3000 and initial conditions (x(0), y(0), z(0)) = (1, 1, 1): (a) hyperchaotic attractors and (b) local amplification of (a)

Grahic Jump Location
Fig. 3

Bifurcation diagram, LEs and x–y phase plane of the time-delayed system (4): (a) bifurcation diagram, (b) local amplification of (a), (c) LEs diagram, (d) x–y phase plane for τ = 2.7 (period-1), (e) x–y phase plane for τ = 2.8 (period-2), and (f) x–y phase plane for τ = 2.9 (period-4)

Grahic Jump Location
Fig. 4

Multiscroll attractors of the nontime-delayed system (3) and the time-delayed system (4) when initial conditions (x(0), y(0), z(0)) = (1, 1, 1): (a) system (3) and (b) system (4) with τ = 0.1

Grahic Jump Location
Fig. 5

Different numbers of scrolls of the time-delayed system (4) when initial conditions (x(0), y(0), z(0)) = (0, 1, 0) and t = 3000: (a) τ = 0.1 and (b) τ = 0.5

Grahic Jump Location
Fig. 6

Coexistence of multiple attractors of the time-delayed system (4) when τ = 0.1 and t = 3000: x varying in the interval [−8, 36] for initial conditions (x(0), y(0), z(0)) = (0, 0, 1) and the interval [−36, 16] for initial conditions (x(0), y(0), z(0)) = (0, 1, 0)

Grahic Jump Location
Fig. 7

The electric circuit realization of the time-delayed system (4)

Grahic Jump Location
Fig. 8

Active first-order all pass filter: R30 = R31 = 2.2 kΩ and C = 10 nF

Grahic Jump Location
Fig. 9

Circuit simulations of hyperchaotic attractors with multiscroll of the time-delayed system (4) when initial conditions (x(0), y(0), z(0)) = (1, 1, 1): (a) hyperchaotic attractors and (b) local amplification of (a)

Grahic Jump Location
Fig. 10

Circuit simulations of multiscroll attractors of the nontime-delayed system (3) and the time-delayed system (4) when initial conditions (x(0), y(0), z(0)) = (1, 1, 1): (a) system (3) and (b) system (4) with τ = 0.1

Grahic Jump Location
Fig. 11

Circuit simulations of different numbers of scrolls of the time-delayed system (4) when initial conditions (x(0), y(0), z(0)) = (0, 1, 0) and t = 3000: (a) τ = 0.1, and (b) τ = 0.5

Grahic Jump Location
Fig. 12

Circuit simulations of coexistence of multiple attractors of the time-delayed system (4) when τ = 0.1 and t = 3000: x varying in the interval [−36, 16] for initial conditions (x(0), y(0), z(0)) = (0, 1, 0) and the interval [−8, 36] for initial conditions (x(0), y(0), z(0)) = (0, 0, 1)

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