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A Simple Method for Generating Mirror Symmetry Composite Multiscroll Chaotic Attractors

Xuenan Peng

School of Physics and Optoelectronic Engineering, Xiangtan University, Xiangtan 411105, P. R. China

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Yicheng Zeng

School of Physics and Optoelectronic Engineering, Xiangtan University, Xiangtan 411105, P. R. China

E-mail Address: yichengz@xtu.edu.cn

Corresponding author.

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https://doi.org/10.1142/S021812742050220XCited by:10 (Source: Crossref)

Abstract

For further increasing the complexity of chaotic attractors, a new method for generating Mirror Symmetry Composite Multiscroll Chaotic Attractors (MSCMCA) is proposed. We take the Lorenz system as an example to explain the mechanism of the method. Firstly, by varying the signs and magnitudes of the nonlinear terms, the Lorenz system generates symmetrical attractors and different-magnitude attractors, respectively. Secondly, a modified Lorenz system is constructed by imposing several unified multilevel-logic pulse signals to the Lorenz system. The new system generates a novel chaotic attractor consisting of two pairs of different-magnitude symmetrical attractors. By adjusting the parameters of the pulse signals, the modified Lorenz system can also be controlled to generate novel grid multiscroll chaotic attractors, namely MSCMCA. Several dynamical behaviors of the new system are shown by equilibria analysis and Lyapunov exponent spectrum. Moreover, the method can be applied to other chaotic systems. Finally, a circuit of the modified Lorenz system is designed by Multisim software, and the simulation result proves the effectiveness of the method.

Keywords:

References

Ai, W., Sun, K. & Fu, Y. [2018] “ Design of multiwing-multiscroll grid compound chaotic system and its circuit implementation,” Int. J. Mod. Phys. C 29, 1850049. Link, Web of Science, Google Scholar
Alpar, O. [2015] “ Three winged lateen shaped chaotic attractor,” Nonlin. Dyn. 82, 435–449. Crossref, Web of Science, Google Scholar
Bhalekar, S. B. [2012] “ Forming mechanizm of Bhalekar–Gejji chaotic dynamical system,” Am. J. Comput. Appl. Math. 87, 257–259. Google Scholar
Bouali, S. [2012] “ A novel strange attractor with a stretched loop,” Nonlin. Dyn. 70, 2375–2381. Crossref, Web of Science, Google Scholar
Campos-Cantón, E. [2016] “ Chaotic attractors based on unstable dissipative systems via third-order differential equation,” Int. J. Mod. Phys. C 27, 1650008. Link, Web of Science, Google Scholar
Cang, S., Wang, Z., Chen, Z. & Jia, H. [2014] “ Analytical and numerical investigation of a new Lorenz-like chaotic attractor with compound structures,” Nonlin. Dyn. 75, 745–760. Crossref, Web of Science, Google Scholar
Chang, D., Li, Z., Wang, M. & Zeng, Y. [2018] “ A novel digital programmable multi-scroll chaotic system and its application in FPGA-based audio secure communication,” AEU — Int. J. Electron. Commun. 88, 20–29. Crossref, Web of Science, Google Scholar
Chen, G. & Ueta, T. [1999] “ Yet another chaotic attractor,” Int. J. Bifurcation and Chaos 9, 1465–1466. Link, Web of Science, Google Scholar
Dantsev, D. [2018] “ A novel type of chaotic attractor for quadratic systems without equilibriums,” Int. J. Bifurcation and Chaos 28, 1850001-1–7. Link, Web of Science, Google Scholar
Elwakil, A. S., Özoǧuz, S. & Kennedy, M. P. [2002] “ Creation of a complex butterfly attractor using a novel Lorenz-type system,” IEEE Trans. Circuits Syst.-I 49, 527–530. Crossref, Web of Science, Google Scholar
Gholizadeh, A., Nik, H. S. & Jajarmi, A. [2015] “ Analysis and control of a three-dimensional autonomous chaotic system,” Appl. Math. Inf. Sci. 727, 739–747. Google Scholar
Guo, Y., Qi, G. & Hamam, Y. [2016] “ A multi-wing spherical chaotic system using fractal process,” Nonlin. Dyn. 85, 2765–2775. Crossref, Web of Science, Google Scholar
Hong, Q., Xie, Q. & Xiao, P. [2017] “ A novel approach for generating multi-direction multi-double-scroll attractors,” Nonlin. Dyn. 87, 1015–1030. Crossref, Web of Science, Google Scholar
Hong, Q., Li, Y., Wang, X. & Zeng, Z. [2018] “ A versatile pulse control method to generate arbitrary multi-direction multi-butterfly chaotic attractors,” IEEE Trans. Comput.-Aided Desig. Integr. Circuits Syst., pp. 1–1. Web of Science, Google Scholar
Hu, G. [2009a] “ Generating hyperchaotic attractors with three positive Lyapunov exponents via state feedback control,” Int. J. Bifurcation and Chaos 19, 651–660. Link, Web of Science, Google Scholar
Hu, G. [2009b] “ Scheme for doubling the number of wings in hyperchaotic attractors,” Acta Phys. Sin. 58, 8139–8145. Crossref, Web of Science, Google Scholar
Jafari, S., Pham, V. T. & Kapitaniak, T. [2016] “ Multiscroll chaotic sea obtained from a simple 3D system without equilibrium,” Int. J. Bifurcation and Chaos 26, 1650031-1–7. Link, Web of Science, Google Scholar
Kengne, J., Negou, A. N. & Njitacke, Z. T. [2017] “ Antimonotonicity, chaos and multiple attractors in a novel autonomous jerk circuit,” Int. J. Bifurcation and Chaos 27, 1750100-1–20. Link, Web of Science, Google Scholar
Li, D. [2008] “ A three-scroll chaotic attractor,” Phys. Lett. A 372, 387–393. Crossref, Web of Science, Google Scholar
Lorenz, E. N. [1963] “ Deterministic nonperiodic flow,” J. Atmos. Sci. 20, 130–141. Crossref, Web of Science, Google Scholar
Lü, J., Chen, G., Cheng, D. & Celikovsky, S. [2002] “ Bridge the gap between the Lorenz system and the Chen system,” Int. J. Bifurcation and Chaos 12, 2917–2926. Link, Web of Science, Google Scholar
Lü, J., Chen, G., Yu, X. & Leung, H. [2004] “ Design and analysis of multiscroll chaotic attractors from saturated function series,” IEEE Trans. Circuits Syst.-I 51, 2476–2490. Crossref, Web of Science, Google Scholar
Lü, J. & Chen, G. [2006] “ Generating multiscroll chaotic attractors: Theories, methods and applications,” Int. J. Bifurcation and Chaos 16, 775–858. Link, Web of Science, Google Scholar
Ontanon-Garcia, L. J. & Campos-Cantón, E. [2013] “ Preservation of a two-wing Lorenz-like attractor with stable equilibria,” J. Franklin Inst. 350, 2867–2880. Crossref, Web of Science, Google Scholar
Ontanon-Garcia, L. J. & Lozoya-Ponce, R. E. [2017] “ Analog electronic implementation of unstable dissipative systems of type I with multi-scrolls displaced along space,” Int. J. Bifurcation and Chaos 27, 1750093-1–15. Link, Web of Science, Google Scholar
Qi, G., Chen, G., Li, S. & Zhang, Y. [2006] “ Four-wing attractors: From pseudo to real,” Int. J. Bifurcation and Chaos 16, 859–885. Link, Web of Science, Google Scholar
Rajagopal, K., Durdu, A., Jafari, S., Uyaroglu, Y., Karthikeyan, A. & Akgul, A. [2019] “ Multiscroll chaotic system with sigmoid nonlinearity and its fractional order form with synchronization application,” Int. J. Non-Lin. Mech. 116, 262–272. Crossref, Web of Science, Google Scholar
Rössler, O. E. [1976] “ An equation for continuous chaos,” Phys. Lett. A 57, 397–398. Crossref, Web of Science, Google Scholar
Slimane, N. B., Bouallegue, K. & Machhout, M. [2017] “ Designing a multi-scroll chaotic system by operating Logistic map with fractal process,” Nonlin. Dyn. 88, 1655–1675. Crossref, Web of Science, Google Scholar
Sprott, J. C. [1994] “ Some simple chaotic flows,” Phys. Rev. E 50, R647. Crossref, Web of Science, Google Scholar
Sprott, J. C. [2014] “ A dynamical system with a strange attractor and invariant tori,” Phys. Lett. A 378, 1361–1363. Crossref, Web of Science, Google Scholar
Sun, C., Chen, Z. & Xu, Q. [2017] “ Generating a double-scroll attractor by connecting a pair of mutual mirror-image attractors via planar switching control,” Int. J. Bifurcation and Chaos 27, 1750197-1–16. Link, Web of Science, Google Scholar
Suykens, J. A. & Vandewalle, J. [1993] “ Generation of n-double scrolls $(n = 1, 2, 3, 4, \dots)$ ,” IEEE Trans. Circuits Syst.-I 40, 861–867. Crossref, Web of Science, Google Scholar
Xiong, L., Zhang, S., Zeng, Y. & Liu, B. [2018] “ Dynamics of a new composite four-Scroll chaotic system,” Chin. J. Phys. 56, 2381–2394. Crossref, Web of Science, Google Scholar
Yalçin, M. E., Suykens, J. A., Vandewalle, J. & Özoǧuz, S. [2002] “ Families of scroll grid attractors,” Int. J. Bifurcation and Chaos 12, 23–41. Link, Web of Science, Google Scholar
Zhang, C. & Yu, S. [2010] “ On constructing complex grid multi-wing hyperchaotic system: Theoretical design and circuit implementation,” Int. J. Circ. Th. Appl. 41, 221–237. Crossref, Web of Science, Google Scholar
Zhang, C., Yu, S. & Chen, G. [2012] “ Design and implementation of compound chaotic attractors,” Int. J. Bifurcation and Chaos 22, 1250120-1–13. Link, Web of Science, Google Scholar
Zhou, J. & Yang, D. [2013] “ Chaos control of a new 3D autonomous system by stability transformation method,” Nonlin. Dyn. 73, 565–577. Crossref, Web of Science, Google Scholar
Zhou, L., Wang, C. & Zhou, L. [2017] “ Generating four-wing hyperchaotic attractor and two-wing, three-wing, and four-wing chaotic attractors in 4D memristive system,” Int. J. Bifurcation and Chaos 27, 1750027-1–14. Link, Web of Science, Google Scholar