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Stability Analysis of Fractional-Order Neural Networks with Time Delay

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Abstract

The stability analysis of fractional-order Hopfield neural networks with time delay is investigated. A stability theorem of fractional-order neural networks with time delay is derived. The stability conditions of the two-dimensional fractional-order neural networks with time delay are obtained. Furthermore, the three-dimensional fractional-order neural networks with different ring structures and time delay are proposed, and their stability conditions are derived. To illustrate the effectiveness of our theoretical results, numerical examples and simulations are also presented.

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References

  1. Koeller RC (1984) Application of fractional calculus to the theory of viscoelasticity. J Appl Mech 51:294–298

    Article  MathSciNet  Google Scholar 

  2. Koeller RC (1986) Polynomial operators, stieltjes convolution, and fractional calculus in hereditary mechanics. Acta Mech 58:251–264

    Article  MathSciNet  MATH  Google Scholar 

  3. Heaviside O (1971) Electromagnetic theory. Chelsea, New York

    Google Scholar 

  4. Podlubny I (1999) Fractional differential equations. Academic Press, New York

    Google Scholar 

  5. Shantanu D (2011) Functional fractional calculus. Springer, Berlin

    Google Scholar 

  6. Li Y, Chen YQ, Podlubny I (2009) Mittag–Leffler stability of fractional order nonlinear dynamic systems. Automatica 45(8):1965–1969

    Article  MathSciNet  MATH  Google Scholar 

  7. Sabatier J, Moze M, Farges C (2010) LMI stability conditions for fractional order systems. Comput Math Appl 59:1594–1609

    Article  MathSciNet  MATH  Google Scholar 

  8. Zhang FR, Li CP, Chen YQ (2011) Asymptotical stability of nonlinear fractional differential system with Caputo derivative. Int J Differ Equ, p 635165

  9. Hadi D, Dumitru B, Jalil S (2012) Stability analysis of Caputo fractional-order nonlinear systems revisited. Nonlinear Dyn 67:2433–2439

    Article  Google Scholar 

  10. Yu JM, Hu H, Zhou SB, Lin XR (2013) Generalized Mittag–Leffler stability of multi-variables fractional order nonlinear systems. Automatica 49:1798–1803

    Article  MathSciNet  MATH  Google Scholar 

  11. Bonnet C, Partington JR (2002) Analysis of fractional delay systems of retarded and neutral type. Automatica 38:1133–1138

    Article  MathSciNet  Google Scholar 

  12. Wang CH, Cheng YC (2006) A numerical algorithm for stability testing of fractional delay systems. Automatica 42:825–831

    Article  MATH  Google Scholar 

  13. Yang ZH, Cao JD (2013) Initial value problems for arbitrary order fractional differential equations with delay. Commun Nonlinear Sci Numer Simul 18:2993–3005

    Article  MathSciNet  Google Scholar 

  14. Deng WH, Li CP, Lü JH (2007) Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dyn 48(4):409–416

    Article  MATH  Google Scholar 

  15. Lazarević M, Spasi A (2009) Finite-time stability analysis of fractional order time delay systems: Gronwall’s approach. Math Comput Model 49:475–481

    Article  Google Scholar 

  16. Afshin M, Mohammad H (2013) Stability of linear time invariant fractional delay systems of retarded type in the space of delay parameters. Automatica 49:1287–1294

    Article  Google Scholar 

  17. Hopfield J (1984) Neurons with graded response have collective computational properties like those of two-state neurons. Proc Natl Acad Sci USA 81:3088–3092

    Article  Google Scholar 

  18. Wang JL, Wu HN, Guo L (2011) Passivity and stability analysis of reaction-diffusion neural networks with dirichlet boundary conditions. IEEE Trans Neural Netw 22(12):2105–2116

    Article  Google Scholar 

  19. Wang JL, Wu HN, Guo L (2013) Stability analysis of reaction-diffusion Cohen–Grossberg neural networks under impulsive control. Neurocomput 106:21–30

    Article  MathSciNet  Google Scholar 

  20. Kao YG, Wang CH, Zhang L (2013) Delay-dependent robust exponential stability of impulsive markovian jumping reaction-diffusion Cohen–Grossberg neural networks. Neural Process Lett 38(3):321–346

    Article  Google Scholar 

  21. Arik S (2004) An analysis of exponential stability of delayed neural networks with time varying delays. Neural Netw 17:1027–1031

    Article  Google Scholar 

  22. Mou S, Gao H, Qiang W, Chen K (2008) New delay-dependent exponetial stability for neural networks with time delay. IEEE Trans Syst Man Cybern B Cybern 38(2):571–576

    Article  Google Scholar 

  23. Hu C, Jiang HJ, Teng ZD (2010) Globally exponential stability for delayed neural networks under impulsive control. Neural Process Lett 31(2):105–127

    Article  Google Scholar 

  24. Chandran R, Balasubramaniam P (2013) Delay dependent exponential stability for fuzzy recurrent neural networks with interval time-varying delay. Neural Process Lett 37(2):147–161

    Article  Google Scholar 

  25. Wang JL, Wu HN (2011) Stability analysis of impulsive parabolic complex networks. Chaos Soliton Fract 44(11):1020–1034

    Article  Google Scholar 

  26. Yu WW, Cao JD, Chen GR (2008) Stability and Hopf bifurcation of a general delayed recurrent nrural network. IEEE Trans Neural Netw 19(5):845–854

    Article  MATH  Google Scholar 

  27. Jiang F, Shen Y (2013) Stability of stochastic \(\theta \)-methods for stochastic delay Hopfield neural networks under regime switching. Neural Process Lett 38(3):433–444

    Article  MathSciNet  MATH  Google Scholar 

  28. Lundstrom B, Higgs M, Spain W, Fairhall A (2008) Fractional differentiation by neocortical pyramidal neurons. Nat Neurosci 11:1335–1342

    Article  MATH  Google Scholar 

  29. Kaslik E, Sivasundaram S (2011) Dynamics of fractional-order neural networks. In: Proceedings of the international conference on neural networks. California, USA, IEEE, pp 611–618

  30. Kaslik E, Sivasundaram S (2012) Nonlinear dynamics and chaos in fractional-order neural networks. Neural Netw 32:245–256

    Article  Google Scholar 

  31. Boroomand A, Menhaj M (2009) Fractional-order Hopfield neural networks. Lect Notes Comput Sci 5506:883–890

    Article  Google Scholar 

  32. Arena P, Fortuna L, Porto D (2000) Chaotic behavior in nonintegerorder cellular neural networks. Phys Rev E 61:777–781

    Article  MATH  Google Scholar 

  33. Zhou S, Li H, Zhu Z (2008) Chaos control and synchronization in a fractional neuron network system. Chaos Soliton Fract 36(4):973–984

    Article  MathSciNet  MATH  Google Scholar 

  34. Çelik V, Demir Y (2010) Chaotic fractional order delayed cellular neural network. In: New trends in nanotechnology and fractional calculus applications, pp 313–320

  35. Huang X, Zhao Z, Wang Z, Li Y (2012) Chaos and hyperchaos in fractional-order cellular neural networks. Neurocomput 94:13–21

    Article  Google Scholar 

  36. Bhalekar S, Varsha D (2011) A predictor-corrector scheme for solving nonlinear delay differential equations of fractional order. J Fract Calc Appl 1(5):1–9

    Google Scholar 

  37. Qin YX, Liu YQ, Wang L, Zheng ZX (1989) Stability of dynamic systems with delays (in Chinese), 2nd edn. Science Press, Beijing

    MATH  Google Scholar 

  38. Hu HY, Wang ZH (2002) Dynamical of control mechanical systems with delayed feed-back. Spring-Verlag, New York

    Book  Google Scholar 

  39. Guo S (2005) Spatio-temporal patterns of nonlinear oscillations in an excitatory ring network with delay. Nonlinearity 18(5):2391–2407

    Article  MathSciNet  MATH  Google Scholar 

  40. Guo S, Huang L (2007) Stability of nonlinear waves in a ring of neurons with delays. J Differ Equ 236:343–374

    Article  MathSciNet  Google Scholar 

  41. Bungay SD, Campbell SA (2007) Patterns of oscillation in a ring of identical cells with delayed coupling. Int J Bifurc Chaos 17(9):3109–3125

    Article  MathSciNet  Google Scholar 

  42. Kaslik E, Balint S (2009) Complex and chaotic dynamics in a discretetime-delayed hopfield neural network with ring architecture. Neural Netw 22:1411–1418

    Article  Google Scholar 

  43. Liao X, Wong K, Wu Z (2001) Bifurcation analysis on a two-neuron system with distributed delays. Phys D 149:123–141

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, People’s Republic of China

    Hu Wang, Yongguang Yu, Guoguang Wen & Shuo Zhang

Authors
  1. Hu Wang
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  2. Yongguang Yu
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  3. Guoguang Wen
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  4. Shuo Zhang
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Corresponding author

Correspondence to Yongguang Yu.

Additional information

Supported by National Nature Science Foundation of China (No. 11371049) and Science Foundation of BJTU (2014YJS132).

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Wang, H., Yu, Y., Wen, G. et al. Stability Analysis of Fractional-Order Neural Networks with Time Delay. Neural Process Lett 42, 479–500 (2015). https://doi.org/10.1007/s11063-014-9368-3

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  • DOI: https://doi.org/10.1007/s11063-014-9368-3

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