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Advances in Mathematical Physics
Volume 2013 (2013), Article ID 657245, 9 pages
http://dx.doi.org/10.1155/2013/657245
Research Article

Nonlinear Dynamics and Chaos in Fractional-Order Hopfield Neural Networks with Delay

1Shandong Key Laboratory of Robotics and Intelligent Technology, College of Information and Electrical Engineering, Shandong University of Science and Technology, Qingdao 266590, China
2College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao 266590, China

Received 4 August 2013; Accepted 30 September 2013

Academic Editor: Changpin Li

Copyright © 2013 Xia Huang 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. I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
  2. P. L. Butzer and U. Westphal, An Introduction to Fractional Calculus, World Scientific, Singapore, 2000.
  3. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Hackensack, NJ, USA, 2001.
  4. M. Nakagava and K. Sorimachi, “Basic characteristic of a fractance device,” IEICE Transactions on Fundamentals of Electronics, vol. 75, no. 12, pp. 1814–1818, 1992.
  5. A. Kiani-B, K. Fallahi, N. Pariz, and H. Leung, “A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 3, pp. 863–879, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  6. T. T. Hartley, C. F. Lorenzo, and H. K. Qammer, “Chaos in a fractional order Chua's system,” IEEE Transactions on Circuits and Systems I, vol. 42, no. 8, pp. 485–490, 1995. View at Publisher · View at Google Scholar · View at Scopus
  7. Y. Yu, H.-X. Li, S. Wang, and J. Yu, “Dynamic analysis of a fractional-order Lorenz chaotic system,” Chaos, Solitons and Fractals, vol. 42, no. 2, pp. 1181–1189, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  8. C. Li and G. Peng, “Chaos in Chen's system with a fractional order,” Chaos, Solitons and Fractals, vol. 22, no. 2, pp. 443–450, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  9. C. Li and G. Chen, “Chaos and hyperchaos in the fractional-order Rössler equations,” Physica A, vol. 341, no. 1–4, pp. 55–61, 2004. View at Publisher · View at Google Scholar · View at Scopus
  10. C. P. Li and Y. T. Ma, “Fractional dynamical system and its linearization theorem,” Nonlinear Dynamics, vol. 71, no. 4, pp. 621–633, 2013. View at Zentralblatt MATH
  11. 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 Scopus
  12. V. Daftardar-Gejji and S. Bhalekar, “Chaos in fractional ordered Liu system,” Computers and Mathematics with Applications, vol. 59, no. 3, pp. 1117–1127, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  13. J. G. Lu, “Chaotic dynamics and synchronization of fractional-order Arneodo's systems,” Chaos, Solitons and Fractals, vol. 26, no. 4, pp. 1125–1133, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  14. W.-C. Chen, “Nonlinear dynamics and chaos in a fractional-order financial system,” Chaos, Solitons and Fractals, vol. 36, no. 5, pp. 1305–1314, 2008. View at Publisher · View at Google Scholar · View at Scopus
  15. M.-F. Danca, “Chaotic behavior of a class of discontinuous dynamical systems of fractional-order,” Nonlinear Dynamics, vol. 60, no. 4, pp. 525–534, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  16. J. H. Park, “On global stability criterion for neural networks with discrete and distributed delays,” Chaos, Solitons and Fractals, vol. 30, no. 4, pp. 897–902, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  17. S. Xu, J. Lam, and D. W. C. Ho, “A new LMI condition for delay-dependent asymptotic stability of delayed Hopfield neural networks,” IEEE Transactions on Circuits and Systems II, vol. 53, no. 3, pp. 230–234, 2006. View at Publisher · View at Google Scholar · View at Scopus
  18. X. Liao, K.-W. Wong, Z. Wu, and G. Chen, “Novel robust stability criteria for interval-delayed Hopfield neural networks,” IEEE Transactions on Circuits and Systems I, vol. 48, no. 11, pp. 1355–1359, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  19. S. Mou, H. Gao, J. Lam, and W. Qiang, “A new criterion of delay-dependent asymptotic stability for Hopfield neural networks with time delay,” IEEE Transactions on Neural Networks, vol. 19, no. 3, pp. 532–535, 2008. View at Publisher · View at Google Scholar · View at Scopus
  20. H. Hu, H. Jiang, and Z. Teng, “The boundedness of high-order Hopfield neural networks with variable delays,” Neurocomputing, vol. 73, no. 13–15, pp. 2589–2596, 2010. View at Publisher · View at Google Scholar · View at Scopus
  21. X. X. Liao, J. Wang, and Z. Zeng, “Global asymptotic stability and global exponential stability of delayed cellular neural networks,” IEEE Transactions on Circuits and Systems II, vol. 52, no. 7, pp. 403–409, 2005. View at Publisher · View at Google Scholar · View at Scopus
  22. J. L. Qiu, “Dynamics of high-order Hopfield neural networks with time delays,” Neurocomputing, vol. 73, no. 4–6, pp. 820–826, 2010. View at Publisher · View at Google Scholar · View at Scopus
  23. H. T. Lu, “Chaotic attractors in delayed neural networks,” Physics Letters A, vol. 298, no. 2-3, pp. 109–116, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  24. Q. Li, X.-S. Yang, and F. Yang, “Hyperchaos in Hopfield-type neural networks,” Neurocomputing, vol. 67, no. 1–4, pp. 275–280, 2005. View at Publisher · View at Google Scholar · View at Scopus
  25. M. Gilli, “Strange attractors in delayed cellular neural networks,” IEEE Transactions on Circuits and Systems I, vol. 40, no. 11, pp. 849–853, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  26. B. N. Lundstrom, M. H. Higgs, W. J. Spain, and A. L. Fairhall, “Fractional differentiation by neocortical pyramidal neurons,” Nature Neuroscience, vol. 11, no. 11, pp. 1335–1342, 2008. View at Publisher · View at Google Scholar · View at Scopus
  27. P. Arena, R. Caponetto, L. Fortuna, and D. Porto, “Bifurcation and chaos in noninteger order cellular neural networks,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 8, no. 7, pp. 1527–1539, 1998. View at Zentralblatt MATH · View at Scopus
  28. I. Petráš, “A note on the fractional-order cellular neural networks,” in Proceedings of the International Joint Conference on Neural Networks, pp. 1021–1024, Sheraton Vancouver Wall Centre Hotel, Vancouver, Canada, July 2006.
  29. P. Arena, L. Fortuna, and D. Porto, “Chaotic behavior in noninteger-order cellular neural networks,” Physical Review E, vol. 61, no. 1, pp. 776–781, 2000. View at Scopus
  30. X. Huang, Z. Zhao, Z. Wang, and Y. Li, “Chaos and hyperchaos in fractional-order cellular neural networks,” Neurocomputing, vol. 94, pp. 13–21, 2012.
  31. E. Kaslik and S. Sivasundaram, “Nonlinear dynamics and chaos in fractional-order neural networks,” Neural Networks, vol. 32, pp. 245–256, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  32. V. Çelik and Y. Demir, “Chaotic fractional order delayed cellular neural network,” New Trends in Nanotechnology and Fractional Calculus Applications, vol. 4, pp. 313–320, 2010. View at Zentralblatt MATH
  33. I. Tejado, S. H. Hosseinnia, B. M. Vinagre, X. Song, and Y. Chen, “Dealing with fractional dynamics of IP network delays,” International Journal of Bifurcation and Chaos, vol. 22, no. 4, Article ID 1250089, 2012.
  34. K. Diethelm, N. J. Ford, and A. D. Freed, “A predictor-corrector approach for the numerical solution of fractional differential equations,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 3–22, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  35. K. Diethelm, N. J. Ford, and A. D. Freed, “Detailed error analysis for a fractional Adams method,” Numerical Algorithms, vol. 36, no. 1, pp. 31–52, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  36. S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, and R. Magin, “Fractional Bloch equation with delay,” Computers and Mathematics with Applications, vol. 61, no. 5, pp. 1355–1365, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  37. S. Abbas, “Existence of solutions to fractional order ordinary and delay differential equations and applications,” Electronic Journal of Differential Equations, vol. 2011, no. 9, pp. 1–11, 2011. View at Zentralblatt MATH · View at Scopus