0
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.

FIGURES IN THIS ARTICLE
<>
Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.

References

Biswas, D. , and Banerjee, T. , 2016, “ A Simple Chaotic and Hyperchaotic Time-Delay System: Design and Electronic Circuit Implementation,” Nonlinear Dyn., 83(4), pp. 2331–2347. [CrossRef]
Banerjee, T. , and Biswas, D. , 2013, “ Theory and Experiment of a First Order Chaotic Delay Dynamical System,” Int. J. Bifurcation Chaos, 23(6), p. 1330020. [CrossRef]
Adhikari, B. M. , Prasad, A. , and Dhamala, M. , 2011, “ Time-Delay-Induced Phase-Transition to Synchrony in Coupled Bursting Neurons,” Chaos, 21(2), p. 023116. [CrossRef] [PubMed]
Banerjee, T. , 2012, “ Single Amplifier Biquad Based Inductor-Free Chua's Circuit,” Nonlinear Dyn., 68(4), pp. 565–573. [CrossRef]
Kye, W. H. , Choi, M. , Kurdoglyan, M. S. , Kim, C. M. , and Park, Y. J. , 2004, “ Synchronization of Chaotic Oscillators Due to Common Delay Time Modulation,” Phys. Rev. E, 70(4), p. 046211. [CrossRef]
Özkaynak, F. , 2014, “ Cryptographically Secure Random Number Generator With Chaotic Additional Input,” Nonlinear Dyn., 78(3), pp. 2015–2020. [CrossRef]
Andò, B. , and Graziani, S. , 2000, Stochastic Resonance: Theory and Applications, Kluwer Academic Publishers, Norwell, MA.
Tamaševicius, A. , Mykolaitis, G. , and Bumeliene, S. , 2006, “ Delayed Feedback Chaotic Oscillator With Improved Spectral Characteristics,” Electron. Lett., 42(13), pp. 736–737. [CrossRef]
Srinivasan, K. , Mohamed, I. R. , Murali, K. , Lakshmanan, M. , and Sinha, S. , 2011, “ Design of Time Delayed Chaotic Circuit With Threshold Controller,” Int. J. Bifurcation Chaos, 21(3), pp. 725–735. [CrossRef]
Tamaševicius, A. , Pyragiene, T. , and Meškauskas, M. , 2007, “ Two Scroll Attractor in a Delay Dynamical System,” Int. J. Bifurcation Chaos, 17(10), pp. 3455–3460. [CrossRef]
Suykens, J. A. K. , and Vandewalle, J. , 1993, “ Generation of n-Double Scrolls (n = 1,2,3,4…),” IEEE Trans. Circuits Syst. I, 40(11), pp. 861–867. [CrossRef]
Lü, J. , Yu, S. , Leung, H. , and Chen, G. , 2006, “ Experimental Verification of Multidirectional Multiscroll Chaotic Attractors,” IEEE Trans. Circuits Syst. I, 53(1), pp. 149–165. [CrossRef]
Tang, W. K. S. , Zhong, G. Q. , Chen, G. , and Man, K. F. , 2001, “ Generation of n-Scroll Attractors Via Sine Function,” IEEE Trans. Circuits Syst. I, 48(11), pp. 1369–1372. [CrossRef]
Yu, S. , Lü, J. , and Chen, G. , 2007, “ A Module-Based and Unified Approach to Chaotic Circuit Design and Its Applications,” Int. J. Bifurcation Chaos, 17(5), pp. 1785–1800. [CrossRef]
Yu, S. , Lü, J. , Leung, H. , and Chen, G. , 2005, “ Design and Implementation of n-Scroll Chaotic Attractors From a General Jerk Circuit,” IEEE Trans. Circuits Syst. I, 52(7), pp. 1459–1476. [CrossRef]
Lin, Y. , Wang, C. , and Zhou, L. , 2016, “ Generation and Implementation of Grid Multiscroll Hyperchaotic Attractors Using CCII+,” Optik, 127(5), pp. 2902–2906. [CrossRef]
Yu, S. , and Tang, W. K. S. , 2009, “ Generation of n × m-Scroll Attractors in a Two-Port RCL Network With Hysteresis Circuit,” Chaos, Solitons Fractals, 39(2), pp. 821–830. [CrossRef]
Sprott, J. C. , 2000, “ Simple Chaotic Systems and Circuits,” Am. J. Phys., 68(8), pp. 758–763. [CrossRef]
Sprott, J. C. , 2000, “ A New Class of Chaotic Circuit,” Phys. Lett. A, 266(1), pp. 19–23. [CrossRef]
Yalçin, M. E. , 2007, “ Multi-Scroll and Hypercube Attractors From a General Jerk Circuit Using Josephson Junctions,” Chaos, Solitons Fractals, 34(5), pp. 1659–1666. [CrossRef]
Ruan, S. , and Wei, J. , 2001, “ On the Zeros of a Third Degree Exponential Polynomial With Applications to a Delayed Model for the Control of Testosterone Secretion,” Math. Med. Biol., 18(1), pp. 41–52. [CrossRef]
Hale, J. , and Lunel, S. M. V. , 1993, Introduction to Functional Differential Equations, Springer Verlag, New York.
Farmer, J. D. , 1982, “ Chaotic Attractors of an Infinite-Dimensional Dynamical System,” Physica D, 4(3), pp. 366–393. [CrossRef]
Benettin, G. , Galgani, L. , Giorgilli, A. , and Strelcyn, J. M. , 1980, “ Lyapunov Characteristic Exponents for Smooth Dynamical Systems and for Hamiltonian Systems; A Method for Computing All of Them. Part 1: Theory,” Meccanica, 15(1), pp. 9–20. [CrossRef]
Shampine, L. F. , and Thompson, S. , 2001, “ Solving DDEs in Matlab,” Appl. Numer. Math., 37(4), pp. 441–458. [CrossRef]
Wang, Q. , Yu, S. , Li, C. , and Lü, J. , 2016, “ Theoretical Design and FPGA-Based Implementation of Higher-Dimensional Digital Chaotic Systems,” IEEE Trans. Circuits Syst. I, 63(3), pp. 401–412. [CrossRef]
Ojoniyi, O. S. , and Njah, A. N. , 2016, “ A 5D Hyperchaotic Sprott B System With Coexisting Hidden Attractors,” Chaos, Solitons Fractals, 87, pp. 172–181. [CrossRef]
Li, C. , and Sprott, J. C. , 2014, “ Coexisting Hidden Attractors in a 4-D Simplified Lorenz System,” Int. J. Bifurcation Chaos, 24(3), p. 1450034. [CrossRef]
Ma, J. , Wu, X. , Chu, R. , and Zhang, L. , 2014, “ Selection of Multi-Scroll Attractors in Jerk Circuits and Their Verification Using Pspice,” Nonlinear Dyn., 76(4), pp. 1951–1962. [CrossRef]

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)

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In