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ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

A New Comparison Theorem and Stability Analysis of Fractional Order Cohen-Grossberg Neural Networks

Discontinuity, Nonlinearity, and Complexity 7(1) (2018) 43--53 | DOI:10.5890/DNC.2018.03.004

Xiaolei Liu, Jian Yuan, Gang Zhou, Wenfei Zhao

School of Basic Sciences for Aviation, Naval Aviation University, Yantai 264001, P.R.China

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Abstract

This paper proposes a new comparison theorem and stability analysis of fractional order Cohen-Grossberg neural networks. Firstly, a new comparison theorem for fractional order systems is proved. Secondly, the stability of a class of fractional order Cohen-Grossberg neural networks with Caputo derivative is investigated on the basis of the above comparison theorem. Thirdly, sufficient conditions of stability of the neural networks are obtained utilizing the property of Mittag-Leffler functions, the generalized Gronwall-Bellman inequality and the method of the integral transform. Furthermore, a numerical simulation example is presented to illustrate the effectiveness of these results.

Acknowledgments

All the authors acknowledge the valuable suggestions from the peer reviewers. This work was supported by the Natural Science Foundation of Shandong Province (Grant No. ZR2014AM006).

References

  1. [1]  Yuan, L.G., Yang, Q.G., and Zeng, C.B. (2013), Chaos detection and Parameter identification in fractional-order chaotic systems with delay, Nonlinear Dynamics, 73, 439-448.
  2. [2]  Tenreiro, M.J., Kiryakova, V., and Mainardi, F. (2011), Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul, 16, 1140-1153.
  3. [3]  Sayevand, K., Golbabai, A., and Yildirim, A. (2011), Analysis of differential equations of fractional order, Applied Mathematical Modelling, 36, 4356-4364.
  4. [4]  Bonnet, C. and Partington, J.R. (2000), Coprime factorizations and stability of fractional differential systems, Syst. Control Lett, 41, 167-174.
  5. [5]  Deng, W.H., Li, C.P., and Lu, J.H. (2007), Stability analysis of linear fractional differential system with multiple time-delays, Nonlinear Dynamic, 48, 409-416.
  6. [6]  Li, C.P. and Deng, W.H. (2007), Remarks on fractional derivatives, Appl. Math. Comput., 187, 777-784.
  7. [7]  Sadati, S.J., Baleanu, D., Ranjbar, A., Ghaderi, R., and Abdeljawad, T. (2010), Mittag-Leffler stability theorem for fractional nonlinear systems with delay, Abstr. Appl. Anal., 108651. Doi:10.1155/2010/108651.
  8. [8]  Liu, L. and Zhong, S. (2011), Finite-time stability analysis of fractional order with multi-state time delay, Word Acad. Sci. , Eng. Technol., 76, 874-877.
  9. [9]  Lazarevic, M. (2007), Finite-time stability analysis of fractional order time delay systems: Bellman-Grronwall’s approach, Sci. Tech. Rev., 7, 8-15.
  10. [10]  Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and Applications of Fractional Differential Equations, Elsevier, B. V.
  11. [11]  Yu, J., Hu, C., and Jiang, H.J. (2012), a-stability and a-synchronization for fractional-order neural networks, Neural Networks, 35, 82-87.
  12. [12]  Delavari, H., Baleanu, D., and Sadati, J. (2012), Stability analysis of Caputo fractional-order nonlinear systems revisited, Nonlinear Dynamics, 67, 2433-2439.
  13. [13]  Zhang, R.X., Tian, G., Yang, S.P., and Cao, H.F. (2015), Stability analysis of a class of fractional order nonlinear systems with order lying in (0, 2), ISA Transactions, http://dx.doi.org/10.1016/j.isatra.2014.12.006.
  14. [14]  Jalilian, Y. and Jalilian, R. (2013), Existence of solution for delay fractional differential equations, Mediterr. J. Math., 10, 1731-1747.
  15. [15]  Chen, L.P., Chai, Y., Wu, R.C., Ma, T.D., and Zhai, H.Z.(2013), Dynamic analysis of a class of fractional order neural networks with delay, Neurocomputing, 111, 190-194.
  16. [16]  Li, Y., Chen, Y.Q., and Podlubny, I. (2009), Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica, 45, 1965-1969.
  17. [17]  Arefeh, B. and Mohammad, B.M. (2009), Fractional-Order Hopfield neural networks[J], M. Koppen et al. (Eds.): ICONIP 2008, Part I, LNCS 5506, 883-890.
  18. [18]  Chen, J.J., Zeng, Z.G., and Jiang, P. (2014), Global Mittag-Leffler stability and synchronization of memristor-based fractional-order neural networks, Neural networks, 51, 1-8.
  19. [19]  Sadati, S.J., Baleanu, D., Ranjber, A., Ghaderi, R., and Abdeljawad, T. (2010), Mittag-Leffler Stability Theorem for Fractional Nonlinear Systems with Delay, Abstract and Applied Analysis, 2010, Article ID108651, 7 pages. Doi:10.1155/2010/108651.
  20. [20]  Yu, J.M., Hu, H., Zhou, S.B., and Lin, X.R. (2013), Generalized Mittag-Leffler stability of multi-variables fractional order nonlinear systems, Automatica, 49, 1798-1803.
  21. [21]  Wu, R.C., Lu, Y.F., and Chen, L.P. (2014), Finite-time stability of fractional delayed neural networks, Neurocomputing, http://dx.Doi.org/10.1016/j.neucom.2014.07.060.
  22. [22]  Bao, H.B. and Cao, J.D., Projective synchronization of fractional-order memristor-based neural networks, Neural Networks, 63, 1-9.
  23. [23]  Dlavari, H., Baleanu, D., and Sadati, J. (2012), Stability analysis of Caputo frational-order nonlinear systems revisited, Nonlinear Dyn., 67, 2433-2439.
  24. [24]  Wang, Z.L., Yang, D.S., and Ma, T.D. (2014), Stability analysis for nonlinear fractional-order systems based on comparison principle, Nonlinear Dyn., 75, 387-402.
  25. [25]  Wang, H., Yu, Y.G., Wen, G.G., Zhang, S., and Yu, J.Z. (2015), Global stability analysis of fractional-order Hopfield neural networks with time delay, Neurocomputing, 154, 15-23.
  26. [26]  Liang, S., Wu, R.C., and Chen. L.P. (2015), Comparison principles and stability of nonlinear fractional-order cellular neural networks with multiple time delays, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2015.05.063.
  27. [27]  Gorenflo, R., Kilbas, A.A.,Mainardi, F., and Rogosin, S.V.,Mittag-Leffler Functions, Related Topics and Applications, Springer.
  28. [28]  Stamova, I.M. (2015) , Mittag-Leffler stability of impulsive differential equations of fractional order, Quarterly of Applied Mathematics, http://dx.doi.org/10.1090/qam/1394.
  29. [29]  Zhang, S., Yu, Y.G., and Wang, H. (2014), Mittag-Leffler stability of fractional-order Hopfield neural networks, Nonlinear Analysis: Hybrid Systems. http://dx.doi.org/10.1016/j.nahs.2014.10.001.
  30. [30]  Ye, H., Gao, J., and Ding, Y. (2007), A generalized Gronwall inequality and its application to a fractional differential equation, J. Math. Anal. Appl., 328, 1075-1081.
  31. [31]  De la Sen, M. (2011), About robust stability of Caputo linear fractional dynamic systems with time delays through fixed point theory, Fixed Point Theory Appl., 867932.
  32. [32]  Liang, Y.J., Chen,W., and Richard, L. (2016), Connecting complexity with spectral entropy using the laplace transformed solution to the fractional diffusion equation, Physica A Statistical Mechanics & Its Applications, 453, 327-335.
  33. [33]  Chen,W. amd Pang, G. (2016), A new definiton of fractional Laplacian with application to modeling three-dimensional nonlocal heat conduction, Journal of Computational Physics, 309, 350-367.
  34. [34]  Liang, Y. and Chen, W. (2015), Argularized Miner’s rule for fatigue reliability analysis with Mittag-Leffler statistics, International Journal of Damage Mechanics, 1056789515607610.

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