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Hyperchaos in SC-CNN based modified canonical Chua’s circuit

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Abstract

In this paper, a state-controlled cellular neural network (SC-CNN)-based hyperchaotic circuit is implemented for classical modified canonical Chua’s circuit. The proposed system is modeled by using a suitable connection of four state-controlled generalized CNN cells, while the stability of the circuit is studied by determining the eigenvalues of the stability matrices, the system parameter is varied, and the dynamics as well as the onset of chaos and hyperchaos followed by a period-three doubling bifurcation has been studied through numerical analysis of the generalized SC-CNN equations and real-time experiments. We further validate our findings, the chaotic and hyperchaotic dynamics, characterized by two positive Lyapunov exponents and Lyapunov dimension, is described by a set of four coupled first-order generalized SC-CNN equations. This has been investigated extensively not only analyzing by computer simulation but also demonstrating by laboratory experiments. The experimental results such as phase portraits, Poincaré surface sections and power spectra are in good agreement with those of numerical computations.

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References

  1. Peng, J.H., Ding, E.J., Ding, M., Yang, W.: Synchronizing hyperchaos with a scalar transmitted signal. Phys. Rev. Lett. 76, 904–907 (1996)

    Article  Google Scholar 

  2. Tamaševičius, A., Namajunas, A., Čenys, A.: Simple 4D chaotic oscillator. Electron. Lett. 32, 957–958 (1996b)

    Article  Google Scholar 

  3. Tamaševičius, A., Čenys, A., Mykolaitis, G., Namajunas, A., Lindberg, E.: Hyperchaotic oscillator with gyraiors. Electron. Lett. 33, 542–544 (1997)

    Article  Google Scholar 

  4. Blakely, J.N., Gauthier, D.J.: Attractor bubbling in coupled hyperchaotic oscillators. Int. J. Bifurcat. Chaos 10, 835–847 (2000)

    Google Scholar 

  5. Thamilmaran, K., Lakshmanan, M., Venkatesan, A.: Hyperchaos in a modified canonical Chua’s circuit. Int. J. Bifurcat. Chaos 14, 221–243 (2004)

    Article  MATH  Google Scholar 

  6. Röossler, O.E.: An equation for hyperchaos. Phys. Lett. A 71, 155–157 (1979)

    Article  MathSciNet  Google Scholar 

  7. Matsumoto, T., Chua, L.O., Kobayashi, K.: Hyperchaos: laboratory experiment and numerical confirmation. IEEE Trans. Circuits Syst. 33, 1143–1147 (1986)

    Article  MathSciNet  Google Scholar 

  8. Mitsubori, K., Saito, T.: A four dimensional plus hysteresis chaos generator. IEEE Trans. Circuits Syst. I 41, 782–789 (1994)

    Article  MATH  Google Scholar 

  9. Saito, T.: The dead-zone conductor hyperchaos generator. Electron. Commun. Jpn. 72, 58–67 (1990)

    Article  Google Scholar 

  10. Stoop, R., Peinke, J., Parisi, J., Rohricht, B., Huebener, R.P.: A p-Gesemiconductor experiment showing chaos and hyperchaos. Physica D 35, 425–435 (1989)

    Article  Google Scholar 

  11. Kapitanik, T., Chua, L.O.: Hyperchaotic attractor of unidirectionally-coupled Chua’s circuits. Int. J. Bifurcat. Chaos 4, 477–482 (1994)

    Article  Google Scholar 

  12. Kapitaniak, T., Chua, L.O., Zhong, G.O.: Experimental hyperchaos in coupled Chua’s circuits. IEEE Trans. Circuits Syst. I 41(41), 499–503 (1994)

    Article  Google Scholar 

  13. Kaneko, K.: Doubling of Torus. Prog. Theor. Phys. 69, 1806–1810 (1983)

    Article  MATH  Google Scholar 

  14. Harrison, M.A., Lai, Y.C.: Route to highdimensional cahos. Phys. Rev. E 59, R3799–R3802 (1999)

    Article  Google Scholar 

  15. Kapitaniak, T., Maistrenko, Y., Popovych, S.: Chaos-hyperchaos transition. Phys. Rev. E 62, 1972–1976 (2000)

    Article  Google Scholar 

  16. Li, G.N., Chen, X.Y.: Study on eigenvalue space of hyperchaotic canonical four-dimensional Chua’s circuit. Chin. Phys. B 19, 030507 (2010)

  17. Ishaq Ahamed, A., Lakshmanan, M.: Nonsmooth bifurcations, transient hyperchaos and hyperchaotic beats in a memristive Murali-Lakshmanan- Chua circuit. Int. J. Bifurcat. Chaos 23, 1350098 (2013)

    Article  MathSciNet  Google Scholar 

  18. Chua, L.O., Yang, L.: Cellular neural networks: theory. IEEE Trans. Circuits Syst. 35, 1257–1272 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  19. Arena, P., Baglio, S., Fortuna, L., Manganaro, G.: Chua’s circuit can be generated by CNN cells. IEEE Trans. Circuits Syst. I 42, 123–125 (1995)

    Article  Google Scholar 

  20. Gunay, E.: MLC circuit in the frame of CNN. Int. J. Bifurcat. Chaos 20, 3267–3274 (2010)

    Article  Google Scholar 

  21. Gunay, E.: A new autonomous chaos generator from state controlled Cellular Neural Networks. Int. J. Bifurcat. Chaos 22, 1250069 (2012)

    Article  Google Scholar 

  22. Kilic, R.: SC-CNN based multifunction signal generator. Int. J. Bifurcat. Chaos 17, 4387–4393 (2007)

    Article  Google Scholar 

  23. Kilic, R., Alci, M.: A SC-CNN based chaotic masking system with feedback. Int. J. Bifurcat. Chaos 14, 245–256 (2004)

    Article  MATH  Google Scholar 

  24. Arena, P., Baglio, S., Fortuna, L., Manganaro, G.: Hyperchaos from cellular neural networks. Electron. Lett. 31, 250–251 (1995)

    Article  Google Scholar 

  25. Caponetto, R., Fortuna, L., Occhipinti, L., Xibilia, M.G.: Hyperchaotic dynamic generation via SC-CNNs for secure transmission applications. IEEE Conf. Proc. 1, 492–496 (1998)

    Google Scholar 

  26. Cafagna, D., Grassi, G.: Two-cell cellular neural networks: generation of new hyperchaotic multiscroll attractors. IEEE Conf. Proc. 2, 924–929 (2003)

    Google Scholar 

  27. Yang, X.-S., Yang, F.: Chaos and hyperchaos in a class of simple cellular neural networks modeled by O.D.E. Int. J. Bifurcat. Chaos 16, 2729–2736 (2006)

    Article  MATH  Google Scholar 

  28. Swathy, P.S., Thamilmaran, K.: An experimental study on SC-CNN based canonical Chua’s Circuit. Nonlinear Dyn. 71, 505–514 (2013)

    Article  MathSciNet  Google Scholar 

  29. Chua, L.O., Lin, G.: Canonical realization of Chua’s circuit family. IEEE Trans. Circuits Syst. 37, 885–902 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  30. Kyprianidis, I.M., Petrani, M.L., Kalomiros, J.A., Anagnostopoulos, A.N.: Crisis-induced intermittency in a third-order electrical circuit. Phys. Rev. E 52, 2268–2273 (1995)

    Article  Google Scholar 

  31. Wu, C.W., Rulkòv, N.F.: Studying chaos via 1-D maps-A tutorial. IEEE Trans. Circuits Syst. I 40, 707–721 (1993)

    Article  MATH  Google Scholar 

  32. Banerjee, S., Grebogi, C.: Border collision bifurcations in two-dimensional piecewise smooth maps. Phys Rev. E 59, 4052–4061 (1999)

    Article  Google Scholar 

  33. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physica D 16, 285–317 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  34. Wu, X.P., Wang, L.: Co-dimension-2 bifurcation of coupled BVP oscillators with hard characteristics. Appl. Math. Comput. 219, 5303–5320 (2013)

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Acknowledgments

PSS acknowledges the University Grants Commission (UGC) for the financial assistance through RFSMS scheme. KT acknowledges DST, Govt. of India for the financial support through the Grant No. SR/S2/HEP-015/2010.

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  1. Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli, 620 024, India

    P. S. Swathy & K. Thamilmaran

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  1. P. S. Swathy
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  2. K. Thamilmaran
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Correspondence to K. Thamilmaran.

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Swathy, P.S., Thamilmaran, K. Hyperchaos in SC-CNN based modified canonical Chua’s circuit. Nonlinear Dyn 78, 2639–2650 (2014). https://doi.org/10.1007/s11071-014-1615-7

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