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Discrete Dynamics in Nature and Society
Volume 2014 (2014), Article ID 529358, 6 pages
http://dx.doi.org/10.1155/2014/529358
Research Article

Delay-Dependent Stability Criterion of Caputo Fractional Neural Networks with Distributed Delay

1Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
2Department of Mathematics, Southeast University, Nanjing 210096, China

Received 18 November 2013; Accepted 26 November 2013; Published 12 January 2014

Academic Editor: Guanghui Wen

Copyright © 2014 Abdulaziz Alofi 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. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. View at MathSciNet
  2. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Technical University of Kosice, Kosice, Slovak Republic, 1999. View at MathSciNet
  3. K. Diethelm, The Analysis of Fractional Differential Equations, vol. 2004, Springer, Berlin, Germany, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  4. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. View at MathSciNet
  5. V. Lakshmikantham, “Theory of fractional functional differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 10, pp. 3337–3343, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 8, pp. 2677–2682, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. M. P. Lazarević, “Finite time stability analysis of PDα fractional control of robotic time-delay systems,” Mechanics Research Communications, vol. 33, no. 2, pp. 269–279, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. M. P. Lazarević and A. M. Spasić, “Finite-time stability analysis of fractional order time-delay systems: Gronwall's approach,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 475–481, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. H. Ye, J. Gao, and Y. Ding, “A generalized Gronwall inequality and its application to a fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 1075–1081, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. H. Zhang, J. Cao, and W. Jiang, “General solution of linear fractional neutral differential difference equations,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 489521, 7 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  11. 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
  12. G. A. Anastassiou, “Fractional neural network approximation,” Computers & Mathematics with Applications, vol. 64, no. 6, pp. 1655–1676, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. 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 Scopus
  14. 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
  15. A. Boroomand and M. B. Menhaj, “Fractional-order Hopfield neural networks,” Lecture Notes in Computer Science, vol. 5506, no. 1, pp. 883–890, 2009. View at Publisher · View at Google Scholar · View at Scopus
  16. 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.
  17. S. Zhou, H. Li, and Z. Zhu, “Chaos control and synchronization in a fractional neuron network system,” Chaos, Solitons and Fractals, vol. 36, no. 4, pp. 973–984, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. Zhou, P. Hu, and H. Li, “Chaotic synchronization of a fractional neuron network system with time-varying delays,” in Proceedings of the International Conference on Communications, Circuits and Systems (ICCCAS '09), pp. 863–867, July 2009. View at Scopus
  19. L. Chen, Y. Chai, R. Wu, T. Ma, and H. Zhai, “Dynamic analysis of a class of fractional-order neural networks with delay,” Neurocomputing, vol. 111, pp. 190–194, 2013.
  20. X. Yang and J. Cao, “Finite-time stochastic synchronization of complex networks,” Applied Mathematical Modelling, vol. 34, no. 11, pp. 3631–3641, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. N. Cai, W. Li, and Y. Jing, “Finite-time generalized synchronization of chaotic systems with different order,” Nonlinear Dynamics, vol. 64, no. 4, pp. 385–393, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  22. M. Xiao, W. X. Zheng, and J. Cao, “Bifurcation and control in a neural network with small and large delays,” Neural Networks, vol. 44, pp. 132–142, 2013.
  23. Z. Wang, J. Cao, G. Chen, and X. Liu, “Synchronization in an array of nonidentical neural networks with leakage delays and impulsive coupling,” Neurocomputing, vol. 111, pp. 177–183, 2013.
  24. X. Yang, J. Cao, and J. Lu, “Synchronization of Markovian coupled neural networks with nonidentical node-delays and random coupling strengths,” IEEE Transactions on Neural Networks and Learning Systems, vol. 23, no. 1, pp. 60–71, 2012.
  25. X. Yang and J. Cao, “Synchronization of discontinuous neural networks with delays via adaptive control,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 147164, 9 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. Y. Wang and J. Cao, “Cluster synchronization in nonlinearly coupled delayed networks of non-identical dynamic systems,” Nonlinear Analysis. Real World Applications, vol. 14, no. 1, pp. 842–851, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet