Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2017, Article ID 2313768, 13 pages
https://doi.org/10.1155/2017/2313768
Research Article

Characteristic Analysis of Fractional-Order 4D Hyperchaotic Memristive Circuit

1School of Physics and Electronics, Central South University, Changsha 410083, China
2School of Information Science and Engineering, Dalian Polytechnic University, Dalian 116034, China

Correspondence should be addressed to Kehui Sun; nc.ude.usc@iuhek

Received 6 March 2017; Accepted 28 May 2017; Published 10 July 2017

Academic Editor: Alessandro Lo Schiavo

Copyright © 2017 Jun Mou 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. L. O. Chua, “Memristor-the missing circuit element,” IEEE Transactions on Circuit Theory, vol. 18, no. 5, pp. 507–519, 1971. View at Publisher · View at Google Scholar
  2. L. O. Chua and S. M. Kang, “Memristive Devices and Systems,” Proceedings of the IEEE, vol. 64, no. 2, pp. 209–223, 1976. View at Publisher · View at Google Scholar · View at Scopus
  3. J. M. Tour and H. Tao, “Electronics: the fourth element,” Nature, vol. 453, no. 7191, pp. 42-43, 2008. View at Publisher · View at Google Scholar · View at Scopus
  4. D. B. Strukov, G. S. Snider, D. R. Stewart, and R. S. Williams, “The missing memristor found,” Nature, vol. 453, pp. 80–83, 2008. View at Publisher · View at Google Scholar · View at Scopus
  5. F. Corinto, A. Ascoli, and M. Gilli, “Nonlinear dynamics of memristor oscillators,” IEEE Transactions on Circuits and Systems. I. Regular Papers, vol. 58, no. 6, pp. 1323–1336, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. L. Teng, H. C. Herbert, and X. Wang, “Chaotic behavior in fractional-order memristor-based simplest chaotic circuit using fourth degree polynomial,” Nonlinear Dynamics, vol. 77, no. 1-2, pp. 231–241, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  7. H. Kim, M. P. Sah, C. Yang, S. Cho, and L. O. Chua, “Memristor emulator for memristor circuit applications,” IEEE Transactions on Circuits and Systems. I. Regular Papers, vol. 59, no. 10, pp. 2422–2431, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. M. Dhamala, Y.-C. Lai, and E. J. Kostelich, “Analyses of transient chaotic time series,” Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, vol. 64, no. 5, Article ID 056207, pp. 056207/1–056207/9, 2001. View at Publisher · View at Google Scholar · View at Scopus
  9. M. Itoh and L. O. Chua, “Memristor oscillators,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 18, no. 11, pp. 3183–3206, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. B. Muthuswamy and L. O. Chua, “Simplest chaotic circuit,” International Journal of Bifurcation and Chaos, vol. 20, no. 5, pp. 1567–1580, 2010. View at Publisher · View at Google Scholar · View at Scopus
  11. B. Muthuswamy, “Implementing memristor based chaotic circuits,” International Journal of Bifurcation and Chaos, vol. 20, no. 5, pp. 1335–1350, 2010. View at Publisher · View at Google Scholar · View at Scopus
  12. B. Muthuswamy and P. Kokate, “Memristor-based chaotic circuits,” IETE Technical Review, vol. 26, no. 6, pp. 415–426, 2009. View at Publisher · View at Google Scholar · View at Scopus
  13. B. C. Bao, Z. Liu, and J. P. Xu, “Steady periodic memristor oscillator with transient chaotic behaviours,” Electronics Letters, vol. 46, no. 3, pp. 228–230, 2010. View at Publisher · View at Google Scholar · View at Scopus
  14. B.-C. Bao, J.-P. Xu, and Z. Liu, “Initial state dependent dynamical behaviors in a memristor based chaotic circuit,” Chinese Physics Letters, vol. 27, no. 7, Article ID 070504, 2010. View at Publisher · View at Google Scholar · View at Scopus
  15. R. Kilic, Practical Guide for Studying Chua’s Circuits, World Scientific, 2010.
  16. W. Xu and X. Yue, “Global analyses of crisis and stochastic bifurcation in the hardening Helmholtz-Duffing oscillator,” Science China Technological Sciences, vol. 53, no. 3, pp. 664–673, 2010. View at Publisher · View at Google Scholar · View at Scopus
  17. B. Bao, T. Jiang, G. Wang, P. Jin, H. Bao, and M. Chen, “Two-memristor-based Chua's hyperchaotic circuit with plane equilibrium and its extreme multistability,” Nonlinear Dynamics, pp. 1–15, 2017. View at Publisher · View at Google Scholar
  18. B. C. Bao, N. Wang, Q. Xu et al., “A simple third-order memristive band pass filter chaotic circuit,” IEEE Transactions on Circuits and Systems II Express Briefs, p. 99, 2016. View at Google Scholar
  19. B. C. Bao, H. Bao, N. Wang et al., “Hidden extreme multistability in memristive hyperchaotic system,” Chaos Solitons and Fractals, vol. 94, pp. 102–111, 2017. View at Google Scholar
  20. H. Wu, B. Bao, Z. Liu, Q. Xu, and P. Jiang, “Chaotic and periodic bursting phenomena in a memristive Wien-bridge oscillator,” Nonlinear Dynamics, vol. 83, no. 1-2, pp. 893–903, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  21. B. Zhang and F. Q. Deng, “Double-compound synchronization of six memristor-based Lorenz systems,” Nonlinear Dynamics, vol. 77, no. 4, pp. 1519–1530, 2014. View at Publisher · View at Google Scholar · View at Scopus
  22. B. Bao, G. Shi, J. Xu, Z. Liu, and S. Pan, “Dynamics analysis of chaotic circuit with two memristors,” Science China Technological Sciences, vol. 54, no. 8, pp. 2180–2187, 2011. View at Publisher · View at Google Scholar · View at Scopus
  23. J. Mou, K. Sun, J. Ruan, and S. He, “A nonlinear circuit with two memcapacitors,” Nonlinear Dynamics, vol. 86, no. 3, pp. 1–10, 2016. View at Publisher · View at Google Scholar · View at Scopus
  24. F. Yang Y, J. Leng L, and D. Li Q, “The 4-dimensional hyperchaotic memristive circuit based on Chua's circuit,” Acta Physica Sinica, vol. 63, no. 8, article 80502, 2014. View at Google Scholar
  25. A. Wu, “Hyperchaos synchronization of memristor oscillator system via combination scheme,” Advances in Difference Equations, 2014:86, 11 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. J. Ruan Y, K. Sun H, and J. Mou, “Memristor-based Lorenz hyper-chaotic system and its circuit implementation,” Acta Physica Sinica, vol. 65, no. 19, article 190502, 2016. View at Google Scholar
  27. J. Sabatier, O. P. Agrawal, and J. A. T. Machado, “Advances in fractional calculus, Theoretical developments and applications in physics and engineering,” Biochemical Journal, vol. 361, no. 1, pp. 97–103, 2007. View at Google Scholar
  28. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  29. V. Rico-Ramirez, J. Martinez-Lizardo, G. A. Iglesias-Silva, S. Hernandez-Castro, and U. M. Diwekar, “A fractional calculus application to biological reactive systems,” Computer Aided Chemical Engineering, vol. 30, no. 4, pp. 1302–1306, 2012. View at Publisher · View at Google Scholar · View at Scopus
  30. I. Petras, “Fractional-order memristor-based Chua's circuit,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 57, no. 12, pp. 975–979, 2011. View at Publisher · View at Google Scholar · View at Scopus
  31. I. Petras, “Chaos in fractional-order population model,” International Journal of Bifurcation and Chaos, vol. 22, no. 4, Article ID 1250072, 2012. View at Publisher · View at Google Scholar
  32. T. Hartley, C. F. Lorenzo, and H. K. Qammer, “Chaos in a fractional order Chua's system,” IEEE Transactions on Circuits and Systems I Fundamental Theory and Applications, vol. 42, no. 8, pp. 485–490, 1995. View at Publisher · View at Google Scholar
  33. D. Cafagna and G. Grassi, “An effective method for detecting chaos in fractional-order systems,” International Journal of Bifurcation and Chaos, vol. 20, no. 3, pp. 669–678, 2010. View at Publisher · View at Google Scholar · View at Scopus
  34. K. Diethelm, “An algorithm for the numerical solution of differential equations of fractional order,” Electronic Transactions on Numerical Analysis, vol. 5, no. 3, pp. 1–6, 1998. View at Google Scholar · View at MathSciNet
  35. G. Adomian, “A new approach to nonlinear partial differential equations,” Journal of Mathematical Analysis and Applications, vol. 102, no. 2, pp. 420–434, 1984. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  36. 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
  37. M. S. Tavazoei and M. Haeri, “Limitations of frequency domain approximation for detecting chaos in fractional order systems,” Nonlinear Analysis, vol. 69, no. 4, pp. 1299–1320, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  38. D. Cafagna and G. Grassi, “Bifurcation and chaos in the fractional-order Chen system via a time-domain approach,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 18, no. 7, pp. 1845–1863, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  39. D. Cafagna and G. Grassi, “Hyperchaos in the fractional-order Roμssler system with lowest-order,” International Journal of Bifurcation and Chaos, vol. 19, no. 1, pp. 339–347, 2011. View at Publisher · View at Google Scholar · View at Scopus
  40. S. B. He, K. H. Sun, and H. H. Wang, “Solution of the fractional-order chaotic system based on Adomain decomposition algorithm and its complexity analysis,” Acta Physica Sinica, vol. 63, no. 3, Article ID 030502, 2014. View at Google Scholar
  41. 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 MathSciNet
  42. E. F. D. Goufo, “Application of the Caputo-Fabrizio fractional derivative without singular kernel to Korteweg-de Vries-Burgers equation,” Mathematical Modelling and Analysis, vol. 21, no. 2, pp. 188–198, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  43. R. Almeida, “A Caputo fractional derivative of a function with respect to another function,” Communications in Nonlinear Science and Numerical Simulation, vol. 44, pp. 460–481, 2017. View at Google Scholar
  44. S. Momani and K. Al-Khaled, “Numerical solutions for systems of fractional differential equations by the decomposition method,” Applied Mathematics and Computation, vol. 162, no. 3, pp. 1351–1365, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  45. V. Daftardar-Gejji and H. Jafari, “Adomian decomposition: a tool for solving a system of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 301, no. 2, pp. 508–518, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  46. N. T. Shawagfeh, “Analytical approximate solutions for nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 131, no. 2-3, pp. 517–529, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  47. R. Gorenflo and F. Mainardi, Fractal and Fractional Calculus in Continuum Mechanics, Springer-Verlag, 1997.
  48. H. Wang, K. Sun, and S. He, “Characteristic analysis and DSP realization of fractional-order simplified Lorenz system based on Adomian decomposition method,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 25, no. 6, Article ID 1550085, 1550085, 13 pages, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  49. H. A. Larrondo, C. M. González, M. T. Martín, A. Plastino, and O. A. Rosso, “Intensive statistical complexity measure of pseudorandom number generators,” Physica A: Statistical Mechanics and its Applications, vol. 356, no. 1, pp. 133–138, 2005. View at Publisher · View at Google Scholar · View at Scopus
  50. M. Borowiec, A. Rysak, D. N. Betts, C. R. Bowen, H. A. Kim, and G. Litak, “Complex response of a bistable laminated plate: multiscale entropy analysis,” European Physical Journal Plus, vol. 129, no. 10, article no. 211, pp. 1–7, 2014. View at Publisher · View at Google Scholar · View at Scopus
  51. K. H. Sun, S. B. He, Y. He, and L. Z. Yin, “Complexity analysis of chaotic pseudo-random sequences based on spectral entropy algorithm,” Acta Physica Sinica, vol. 62, no. 1, pp. 709–712, 2013. View at Google Scholar
  52. K.-H. Sun, S.-B. He, C.-X. Zhu, and Y. He, “Analysis of chaotic complexity characteristics based on C0 algorithm,” Acta Electronica Sinica, vol. 41, no. 9, pp. 1765–1771, 2013. View at Publisher · View at Google Scholar · View at Scopus