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Topological horseshoe analysis on a four-wing chaotic attractor and its FPGA implement

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Abstract

For a three-dimensional autonomous four-wing chaotic attractor, this paper rigorously verifies its chaotic properties by using topological horseshoe theory and numerical calculations. Firstly, an appropriate Poincaré section of the chaotic attractor is selected by numerical analysis. Accordingly, a certain first return Poincaré map is defined in the Poincaré section. Thereafter, by utilizing numerical calculations and topological horseshoe theory, a one-dimensional tensile topological horseshoe in the Poincaré section is discovered, which revealed that the four-wing attractor has a positive topological entropy, and verifies the existence of chaos in this four-wing attractor. Finally, by using a FPGA chip, the four-wing chaotic attractor was physically implemented, which is more suitable for engineering applications.

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References

  1. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)

    Article  Google Scholar 

  2. Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64, 1196–1199 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Farshidianfar, A., Saghafi, A.: Identification and control of chaos in nonlinear gear dynamic systems using Melnikov analysis. Phys. Lett. A. 378, 3457–3463 (2014)

    Article  MathSciNet  Google Scholar 

  4. Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifur. Chaos. 9, 1465–1466 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Lü, J.H., Chen, G.R., Cheng, D.Z., Celikovsky, S.: Bridge the gap between the Lorenz system and the Chen system. Int. J. Bifur. Chaos. 12, 2917–2926 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Marat, A., Mehmet, O.F.: Generation of cyclic/toroidal chaos by Hopfield neural networks. Neurocomputing. 145, 230–239 (2014)

    Article  Google Scholar 

  7. de la Fraga, L.G., Tlelo-Cuautle, E.: Optimizing the maximum Lyapunov exponent and phase space portraits in multi-scroll chaotic oscillators. Nonlinear Dyn. 76, 1503–1515 (2014)

    Article  Google Scholar 

  8. Soriano-Sánchez, A.G., Posadas-Castillo, C., Platas-Garza, M.A., Diaz-Romero, D.A.: Performance improvement of chaotic encryption via energy and frequency location criteria. Math. Comput. Sim. 112, 14–27 (2015)

    Article  MathSciNet  Google Scholar 

  9. Zhou, Y.C., Bao, L., Chen, C.L.P.: A new 1D chaotic system for image encryption. Signal Process. 97, 172–182 (2014)

    Article  Google Scholar 

  10. Qi, G.Y., Montodo, S.: Hyper-chaos encryption using convolutional masking and model free unmasking. Chin. Phys. B. 23, 050507–1–050507–6 (2014)

    Google Scholar 

  11. Chang, Y., Chen, G.R.: Complex dynamics in Chen’s system. Chaos Solition. Fract. 27, 75–86 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Saptarshi, D., Anish, A., Indranil, P.: Simulation studies on the design of optimum PID controllers to suppress chaotic oscillations in a family of Lorenz-like multi-wing attractors. Math. Comput. Sim. 100, 72–87 (2014)

    Article  MathSciNet  Google Scholar 

  13. Qi, G.Y., van Wyk, B.J., van Wyk, M.A.: A four-wing attractor and its analysis. Chaos Solition. Fract. 40, 2016–2030 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bouallegue, K., Abdessattar, C., Toumi, A.: Multi-scroll and multi-wing chaotic attractor generated with Julia process fractal. Chaos Solition. Fract. 44, 79–85 (2011)

    Article  MathSciNet  Google Scholar 

  15. Zhang, C.X., Yu, S.M.: Generation of grid multi-scroll chaotic attractors via switching piecewise linear controller. Phys. Lett. A. 374(30), 3029–3037 (2010)

    Article  MATH  Google Scholar 

  16. Yu, S.M., Lü, J.H., Chen, G.R.: A family of n-scroll hyperchaotic attractors and their realization. Phys. Lett. A. 364, 244–251 (2007)

    Article  Google Scholar 

  17. Li, Y.X., Tang, W.K.S., Chen, G.R.: Hyperchaos evolved from the generalized Lorenz equation. Int. J. Circuit Theory Appl. 33, 235–251 (2005)

    Article  MATH  Google Scholar 

  18. Wang, J.Z., Chen, Z.Q., Chen, G.R., Yuan, Z.Z.: A novel hyperchaotic system and its complex dynamics. Int. J. Bifur. Chaos. 18, 3309–3324 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu, C.X., Liu, L.: A novel four-dimensional autonomous hyperchaotic system. Chin. Phy. B. 18, 2188–2193 (2009)

    Article  Google Scholar 

  20. Li, Q.D., Tang, S., Yang, X.S.: Hyperchaotic set in continuous chaos-hyperchaos transition. Commun. Nonlinear. Sci. Numer. Simulat. 19, 3718–3734 (2014)

    Article  MathSciNet  Google Scholar 

  21. Huang, X., Zhao, Z., Wang, Z., Li, Y.X.: Chaos and hyperchaos in fractional-order cellular neural networks. Neurocomputing. 94, 13–21 (2012)

    Article  Google Scholar 

  22. Zhou, P., Huang, K.: A new 4-D non-equilibrium fractional-order chaotic system and its circuit implementation. Commun. Nonlinear. Sci. Numer. Simulat. 19, 2005–2011 (2014)

    Article  MathSciNet  Google Scholar 

  23. El-Sayed, A.M.A., Nour, H.M., Elsaid, A., Matouk, A.E., Elsonbaty, A.: Circuit realization, bifurcations, chaos and hyperchaos in a new 4D system. Appl. Math. Comput. 239, 333–345 (2014)

    Article  MathSciNet  Google Scholar 

  24. Trejo-Guerra, R., Tlelo-Cuautle, E., Jiménez-Fuentes, J.M., Sánchez-López, C., Muñoz-Pacheco, J.M., Espinosa-Flores-Verdad, G., Rocha-Pérez, J.M.: Integrated circuit generating 3- and 5-Scroll attractors. Commun. Nonlinear. Sci. Numer. Simulat. 17, 4328–4335 (2012)

    Article  MathSciNet  Google Scholar 

  25. Valtierra-Sánchez de la Vega, J. L., Tlelo-Cuautle, E.: Simulation of piecewise-linear one-dimensional chaotic maps by Verilog-A. IETE Tech. Rev. doi:10.1080/02564602.2015.1018349 (2015)

  26. Wang, G.Y., Bao, X.L., Wang, Z.L.: Design and FPGA Implementation of a new hyperchaotic system. Chin. Phys. B. 17, 3596–3602 (2008)

    Article  Google Scholar 

  27. Tlelo-Cuautle, E., Rangel-Magdaleno, J.J., Pano-Azucena, A.D., Obeso-Rodelo, P.J., Nuñez-Perez, J.C.: FPGA realization of multi-scroll chaotic oscillators. Commun. Nonlinear. Sci. Numer. Simulat. 27, 66–80 (2015)

  28. Yang, X.S., Li, Q.D.: A computer-assisted proof of chaos in Josephson junctions. Chaos Soliton. Fract. 27, 25–30 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wu, W.J., Chen, Z.Q., Yuan, Z.Z.: A computer-assisted proof for the existence of horseshoe in a novel chaotic system. Chaos Soliton. Fract. 41, 2756–2761 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhou, P., Yang, F.Y.: Hyperchaos, chaos, and horseshoe in a 4D nonlinear system with an infinite number of equilibrium points. Nonlinear Dyn. 76, 473–480 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Yang, X.S.: Metric horseshoes. Chaos Soliton. Fract. 20, 1149–1156 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yang, X.S., Tang, Y.: Horseshoes in piecewise continuous maps. Chaos Soliton. Fract. 19, 841–845 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  33. Li, Q.D., Yang, X.S.: A simple method for finding topological horseshoes. Int. J. Bifur. Chaos. 20, 467–478 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Smale, S.: Differentiable dynamical systems. B. Am. Math. Soc. 73, 747–817 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  35. Kennedy, J., Yorke, J.: Topological horseshoes. Trans. Am. Math. Soc. 353, 2513–2530 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  36. Jia, H.Y., Chen, Z.Q., Qi, G.Y.: Topological horseshoe analysis and circuit realization for a fractional-order Lü system. Nonlinear Dyn. 74, 203–212 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ma, C., Wang, X.: Hopf bifurcation and topological horseshoe of a novel finance chaotic system. Commun. Nonlinear. Sci. Numer. Simulat. 17, 721–730 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  38. Li, Q., Huang, S., Tang, S., Zeng, G.: Hyperchaos and horseshoe in a 4D memristive system with a line of equilibria and its implementation. Int. J. Circ. Theor. App. 42, 1172–1188 (2014)

    Article  Google Scholar 

  39. Dong, E.Z., Chen, Z.P., Chen, Z.Q., Yuan, Z.Z.: A novel four-wing chaotic attractor generated from a three-dimensional quadratic autonomous system. Chin. Phys. B. 18, 2680–2689 (2009)

    Article  Google Scholar 

  40. Qi, G., Wyk, M.A., Wyk, B.J., Chen, G.: On a new hyperchaotic system. Phys. Lett. A. 372, 124–136 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  41. Qi, G., Wyk, M.A., Wyk, B.J., Chen, G.: A new hyperchaotic system and its circuit implementation. Chaos, Solit. Fract. 40, 2544–2549 (2009)

    Article  Google Scholar 

Download references

Acknowledgments

This work was partially supported by the Natural Science Foundation of China under Grant Nos. 61203138 and 61374169, the Development of Science and Technology Foundation of the Higher Education Institutions of Tianjin under Grant No. 20120829, the Science and Technology Talent and Technology Innovation Foundation of Tianjin, China, Grant No. 20130830, the Second Level Candidates of 131 Innovative Talents Training Project of Tianjin, China, Grant No. 20130115.

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Authors and Affiliations

  1. Tianjin Key Laboratory For Control Theory and Applications in Complicated Systems, Tianjin University of Technology, Tianjin, 300384, China

    Enzeng Dong & Zhihan Liang

  2. Department of Mechanical Engineering, Tshwane University of Technology, Pretoria, 0001, South Africa

    Shengzhi Du

  3. Department of Automation, Nankai University, Tianjin, 300071, China

    Zengqiang Chen

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  1. Enzeng Dong
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  2. Zhihan Liang
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  4. Zengqiang Chen
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Correspondence to Enzeng Dong.

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Dong, E., Liang, Z., Du, S. et al. Topological horseshoe analysis on a four-wing chaotic attractor and its FPGA implement. Nonlinear Dyn 83, 623–630 (2016). https://doi.org/10.1007/s11071-015-2352-2

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  • DOI: https://doi.org/10.1007/s11071-015-2352-2

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