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Complexity
Volume 2017 (2017), Article ID 8979408, 16 pages
https://doi.org/10.1155/2017/8979408
Research Article

Hyperchaotic Chameleon: Fractional Order FPGA Implementation

Centre for Non-Linear Dynamics, Defense University, Bishoftu, Ethiopia

Correspondence should be addressed to Karthikeyan Rajagopal; moc.liamg@nayekeihtrakr

Received 9 January 2017; Revised 2 March 2017; Accepted 2 March 2017; Published 30 May 2017

Academic Editor: Abdelalim Elsadany

Copyright © 2017 Karthikeyan Rajagopal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. G. A. Leonov, N. V. Kuznetsov, and V. I. Vagaitsev, “Localization of hidden Chua's attractors,” Physics Letters A, vol. 375, no. 23, pp. 2230–2233, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. G. A. Leonov, N. V. Kuznetsov, and V. I. Vagaitsev, “Hidden attractor in smooth Chua systems,” Physica D. Nonlinear Phenomena, vol. 241, no. 18, pp. 1482–1486, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. G. A. Leonov and N. V. Kuznetsov, “Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits,” International Journal of Bifurcation and Chaos, vol. 23, no. 01, Article ID 1330002, 2013. View at Publisher · View at Google Scholar
  4. G. A. Leonov, N. V. Kuznetsov, and T. N. Mokaev, “Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity,” Communications in Nonlinear Science and Numerical Simulation, vol. 28, no. 1–3, pp. 166–174, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. G. A. Leonov, N. V. Kuznetsov, and T. N. Mokaev, “Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion,” The European Physical Journal Special Topics, vol. 224, no. 8, pp. 1421–1458, 2015. View at Publisher · View at Google Scholar · View at Scopus
  6. P. R. Sharma, M. D. Shrimali, A. Prasad, N. V. Kuznetsov, and G. A. Leonov, “Control of multistability in hidden attractors,” The European Physical Journal: Special Topics, vol. 224, no. 8, pp. 1485–1491, 2015. View at Publisher · View at Google Scholar · View at Scopus
  7. P. R. Sharma, M. D. Shrimali, A. Prasad, N. V. Kuznetsov, and G. A. Leonov, “Controlling dynamics of hidden attractors,” International Journal of Bifurcation and Chaos, vol. 25, no. 4, Article ID 1550061, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. S. Jafari, J. C. Sprott, V.-T. Pham, S. M. Hashemi Golpayegani, and A. Homayoun Jafari, “A new cost function for parameter estimation of chaotic systems using return maps as fingerprints,” International Journal of Bifurcation and Chaos, vol. 24, no. 10, Article ID 1450134, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. V.-T. Pham, C. Volos, S. Jafari, Z. Wei, and X. Wang, “Constructing a novel no-equilibrium chaotic system,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 24, no. 5, Article ID 1450073, 6 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. F. R. Tahir, S. Jafari, V.-T. Pham, C. Volos, and X. Wang, “A novel no-equilibrium chaotic system with multiwing butterfly attractors,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 25, no. 4, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. S. Jafari, V.-T. Pham, and T. Kapitaniak, “Multiscroll chaotic sea obtained from a simple 3D system without equilibrium,” International Journal of Bifurcation and Chaos, vol. 26, no. 2, Article ID 1650031, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. S.-K. Lao, Y. Shekofteh, S. Jafari, and J. C. Sprott, “Cost function based on Gaussian mixture model for parameter estimation of a chaotic circuit with a hidden attractor,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 24, no. 1, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. S. Jafari, V.-T. Pham, and T. Kapitaniak, “Multiscroll chaotic sea obtained from a simple 3D system without equilibrium,” International Journal of Bifurcation and Chaos, vol. 26, no. 2, Article ID 1650031, 2016. View at Publisher · View at Google Scholar · View at Scopus
  14. V.-T. Pham, S. Vaidyanathan, C. Volos, S. Jafari, and S. T. Kingni, “A no-equilibrium hyperchaotic system with a cubic nonlinear term,” Optik, vol. 127, no. 6, pp. 3259–3265, 2016. View at Publisher · View at Google Scholar · View at Scopus
  15. V.-T. Pham, S. Vaidyanathan, C. K. Volos, S. Jafari, N. V. Kuznetsov, and T. M. Hoang, “A novel memristive time–delay chaotic system without equilibrium points,” European Physical Journal: Special Topics, vol. 225, no. 1, pp. 127–136, 2016. View at Publisher · View at Google Scholar · View at Scopus
  16. M. Molaie, S. Jafari, J. C. Sprott, and S. M. R. H. Golpayegani, “Simple chaotic flows with one stable equilibrium,” International Journal of Bifurcation and Chaos, vol. 23, no. 11, Article ID 1350188, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. S. T. Kingni, S. Jafari, H. Simo, and P. Woafo, “Three-dimensional chaotic autonomous system with only one stable equilibrium: analysis, circuit design, parameter estimation, control, synchronization and its fractional-order form,” The European Physical Journal Plus, vol. 129, no. 5, pp. 1–16, 2014. View at Publisher · View at Google Scholar · View at Scopus
  18. M. Molaie, S. Jafari, J. C. Sprott, and S. M. Hashemi Golpayegani, “Simple chaotic flows with one stable equilibrium,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 23, no. 11, Article ID 1350188, 7 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. S. T. Kingni, S. Jafari, H. Simo, and P. Woafo, “Three-dimensional chaotic autonomous system with only one stable equilibrium: analysis, circuit design, parameter estimation, control, synchronization and its fractional-order form,” The European Physical Journal Plus, vol. 129, no. 5, article 76, pp. 1–16, 2014. View at Publisher · View at Google Scholar · View at Scopus
  20. V. Pham, S. Jafari, C. Volos, A. Giakoumis, S. Vaidyanathan, and T. Kapitaniak, “A chaotic system with equilibria located on the rounded square loop and its circuit implementation,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 63, no. 9, pp. 878–882, 2016. View at Publisher · View at Google Scholar
  21. R. Karthikeyan, A. Prasina, R. Babu, and S. Raghavendran, “FPGA implementation of novel synchronization methodology for a new chaotic system,” Indian Journal of Science and Technology, vol. 8, no. 11, 2015. View at Publisher · View at Google Scholar · View at Scopus
  22. K. Rajagopal, A. Karthikeyan, and A. K. Srinivasan, “FPGA implementation of novel fractional-order chaotic systems with two equilibriums and no equilibrium and its adaptive sliding mode synchronization,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 87, no. 4, pp. 2281–2304, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  23. D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific, Singapore, 2014.
  24. Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  25. K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, Germany, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  26. M. Pourmahmood Aghababa, “Robust finite-time stabilization of fractional-order chaotic systems based on fractional lyapunov stability theory,” Journal of Computational and Nonlinear Dynamics, vol. 7, no. 2, Article ID 021010, 2012. View at Publisher · View at Google Scholar · View at Scopus
  27. E. A. Boroujeni and H. R. Momeni, “Non-fragile nonlinear fractional order observer design for a class of nonlinear fractional order systems,” Signal Processing, vol. 92, no. 10, pp. 2365–2370, 2012. View at Publisher · View at Google Scholar · View at Scopus
  28. R. Zhang and J. Gong, “Synchronization of the fractional-order chaotic system via adaptive observer,” Systems Science & Control Engineering, vol. 2, no. 1, pp. 751–754, 2014. View at Google Scholar
  29. R. H. Li and W. S. Chen, “Complex dynamical behavior and chaos control in fractional-order Lorenz-like systems,” Chinese Physics B, vol. 22, no. 4, Article ID 040503, 2013. View at Publisher · View at Google Scholar
  30. D. Cafagna and G. Grassi, “Fractional-order systems without equilibria: the first example of hyperchaos and its application to synchronization,” Chinese Physics B, vol. 24, no. 8, Article ID 080502, 2015. View at Publisher · View at Google Scholar · View at Scopus
  31. M.-F. Danca, W. K. Tang, and G. Chen, “Suppressing chaos in a simplest autonomous memristor-based circuit of fractional order by periodic impulses,” Chaos, Solitons & Fractals, vol. 84, pp. 31–40, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  32. I. Petras, “Methos for simulation of the fractional order chaotic systems,” Acta Montanastica Slovaca, vol. 11, no. 4, pp. 273–277, 2006. View at Google Scholar
  33. W. Trzaska Zdzislaw, Matlab Solutions of Chaotic Fractional Order Circuits, InTech, Rijeka, Croatia, 2013, http://www.intechopen.com/download/pdf/21404.
  34. M. A. Jafari, E. Mliki, A. Akgul et al., “Chameleon: the most hidden chaotic flow,” Nonlinear Dynamics, pp. 1–15, 2017. View at Publisher · View at Google Scholar
  35. Q. Li, H. Zeng, and J. Li, “Hyperchaos in a 4D memristive circuit with infinitely many stable equilibria,” Nonlinear Dynamics, vol. 79, no. 4, pp. 2295–2308, 2015. View at Publisher · View at Google Scholar · View at Scopus
  36. Q.-H. Hong, Y.-C. Zeng, and Z.-J. Li, “Design and simulation of chaotic circuit for flux-controlled memristor and charge-controlled memristor,” Acta Physica Sinica, vol. 62, no. 23, Article ID 230502, 2013. View at Publisher · View at Google Scholar · View at Scopus
  37. S. Sampath, S. Vaidyanathan, and V.-T. Pham, “A novel 4-D hyperchaotic system with three quadratic nonlinearities, its adaptive control and circuit simulation,” International Journal of Control Theory and Applications, vol. 9, no. 1, pp. 339–356, 2016. View at Google Scholar · View at Scopus
  38. V. Rashtchi and M. Nourazar, “FPGA implementation of a real-time weak signal detector using a duffing oscillator,” Circuits, Systems, and Signal Processing, vol. 34, no. 10, pp. 3101–3119, 2015. View at Publisher · View at Google Scholar · View at Scopus
  39. E. Tlelo-Cuautle, J. J. Rangel-Magdaleno, A. D. Pano-Azucena, P. J. Obeso-Rodelo, and J. C. Nunez-Perez, “FPGA realization of multi-scroll chaotic oscillators,” Communications in Nonlinear Science and Numerical Simulation, vol. 27, no. 1–3, pp. 66–80, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  40. C. Li, Z. Gong, D. Qian, and Y. Chen, “On the bound of the Lyapunov exponents for the fractional differential systems,” Chaos, vol. 20, no. 1, Article ID 013127, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  41. M. S. Tavazoei and M. Haeri, “Unreliability of frequency-domain approximation in recognising chaos in fractional-order systems,” IET Signal Processing, vol. 1, no. 4, pp. 171–181, 2007. View at Publisher · View at Google Scholar · View at Scopus
  42. B. Bao, P. Jiang, H. Wu, and F. Hu, “Complex transient dynamics in periodically forced memristive Chua’s circuit,” Nonlinear Dynamics, vol. 79, no. 4, pp. 2333–2343, 2015. View at Publisher · View at Google Scholar · View at Scopus
  43. C. Pezeshki, S. Elgar, and R. C. Krishna, “Bispectral analysis of systems possessing chaotic motion,” Journal of Sound and Vibration, vol. 137, no. 3, pp. 357–368, 1990. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  44. D. Dudkowski, S. Jafari, T. Kapitaniak, N. V. Kuznetsov, G. A. Leonov, and A. Prasad, “Hidden attractors in dynamical systems,” Physics Reports. A Review Section of Physics Letters, vol. 637, pp. 1–50, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  45. E. Tlelo-Cuautle, A. D. Pano-Azucena, J. J. Rangel-Magdaleno, V. H. Carbajal-Gomez, and G. Rodriguez-Gomez, “Generating a 50-scroll chaotic attractor at 66 MHz by using FPGAs,” Nonlinear Dynamics, vol. 85, no. 4, pp. 2143–2157, 2016. View at Publisher · View at Google Scholar · View at Scopus
  46. Q. Wang, S. Yu, C. Li et al., “Theoretical design and FPGA-based implementation of higher-dimensional digital chaotic systems,” IEEE Transactions on Circuits and Systems I-Regular Papers, vol. 63, no. 3, pp. 401–412, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  47. E. Dong, Z. Liang, S. Du, and Z. Chen, “Topological horseshoe analysis on a four-wing chaotic attractor and its FPGA implement,” Nonlinear Dynamics, vol. 83, no. 1-2, pp. 623–630, 2016. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  48. E. Tlelo-Cuautle, V. H. Carbajal-Gomez, P. J. Obeso-Rodelo, J. J. Rangel-Magdaleno, and J. C. Núñez-Pérez, “FPGA realization of a chaotic communication system applied to image processing,” Nonlinear Dynamics, vol. 82, no. 4, pp. 1879–1892, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  49. Y.-M. Xu, L.-D. Wang, and S.-K. Duan, “A memristor-based chaotic system and its field programmable gate array implementation,” Wuli Xuebao/Acta Physica Sinica, vol. 65, no. 12, Article ID 120503, 2016. View at Publisher · View at Google Scholar · View at Scopus