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Stability Analysis of Fractional-Order Hopfield Neural Networks with Time-Varying External Inputs

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Abstract

Fractional-order Hopfield neural networks are frequently utilized to model the information processing of neuronal interactions. It is needed to analyze the stability of such neural systems, when their external inputs are time-varying. In this manuscript, some sufficient conditions are presented firstly for the stability of the non-autonomous fractional-order systems by employing Lyapunov functionals. In further, under time-varying external inputs, fractional-order Hopfield neural networks are regards as a class of non-autonomous fractional-order systems, whose general result is used to acquire the stability conditions of our studied neural system. Moreover, a novel robust synchronization method between such neural systems is proposed with the help of the obtained results. Furthermore, some numerical examples are provided to demonstrate the effectiveness of the presented theoretical results.

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References

  1. Podlubny I (1999) Fractiorlal differential eqnations. Academic Press, San Diego

    Google Scholar 

  2. Tripathi D, Pandey SK, Das S (2010) Peristaltic flow of viscoelastic fluid with fractional Maxwell model through a channel. Appl Math Comput 215(10):3645–3654

    MathSciNet  MATH  Google Scholar 

  3. Magin RL, Ovadia M (2008) Modeling the cardiac tissue electrode interface using fractional calculus. J Vib Control 14(9–10):1431–1442

    Article  MATH  Google Scholar 

  4. Magin RL (2010) Fractional calculus models of complex dynamics in biological tissues. Comput Math Appl 59(5):1586–1593

    Article  MathSciNet  MATH  Google Scholar 

  5. Gökdoğan A, Merdan M, Yildirim A (2012) A multistage differential transformation method for approximate solution of Hantavirus infection model. Commun Nonlinear Sci Numer Simul 17(1):1–8

    Article  MathSciNet  Google Scholar 

  6. Weinberg SH (2015) Membrane capacitive memory alters spiking in neurons described by the fractional-Order Hodgkin–Huxley model. PloS One 10(5):e0126629

    Article  Google Scholar 

  7. Xin B, Zhang J (2014) Finite-time stabilizing a fractional-order chaotic financial system with market confidence. Nonlinear Dyn 79(2):1399–1409

    Article  MathSciNet  MATH  Google Scholar 

  8. Boroomand A, Menhaj MB (2008) Fractional-order Hopfield neural networks. In: Koppen M, Kasabov N, Coghill G (eds) Advances in neuro-information processing. Springer, Berlin, pp 883–890

    Google Scholar 

  9. Song C, Cao J (2014) Dynamics in fractional-order neural networks. Neurocomputing 142:494–498

    Article  Google Scholar 

  10. Chen L, Chai Y, Wu R et al (2013) Dynamic analysis of a class of fractional-order neural networks with delay. Neurocomputing 111:190–194

    Article  Google Scholar 

  11. Kaslik E, Sivasundaram S (2011) Dynamics of fractional-order neural networks. In: The 2011 international joint conference on neural networks (IJCNN), IEEE, pp 611–618

  12. Wang H, Yu Y, Wen G (2014) Stability analysis of fractional-order Hopfield neural networks with time delays. Neural Netw 55:98–109

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  14. Chen L, Wu R, Cao J et al (2015) Stability and synchronization of memristor-based fractional-order delayed neural networks. Neural Netw 71:37–44

    Article  Google Scholar 

  15. Yu J, Hu C, Jiang H (2012) \(\alpha \)-stability and \(\alpha \)-synchronization for fractional-order neural networks. Neural Netw 35:82–87

    Article  MATH  Google Scholar 

  16. Chen J, Zeng Z, Jiang P (2014) Global Mittag-Leffler stability and synchronization of memristor-based fractional-order neural networks. Neural Netw 51:1–8

    Article  MATH  Google Scholar 

  17. Li Y, Chen YQ, Podlubny I (2010) Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput Math Appl 59(5):1810–1821

    Article  MathSciNet  MATH  Google Scholar 

  18. Zhang R, Yang S (2013) Robust synchronization of two different fractional-order chaotic systems with unknown parameters using adaptive sliding mode approach. Nonlinear Dyn 71(1–2):269–278

    Article  MathSciNet  Google Scholar 

  19. Li C, Xiong J, Li W et al (2013) Robust synchronization for a class of fractional-order dynamical system via linear state variable. Indian J Phys 87(7):673–678

    Article  MathSciNet  Google Scholar 

  20. Zhang L, Yan Y (2014) Robust synchronization of two different uncertain fractional-order chaotic systems via adaptive sliding mode control. Nonlinear Dyn 76(3):1761–1767

    Article  MathSciNet  MATH  Google Scholar 

  21. Kilbas AAA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier Science Limited, Amersterdam

    MATH  Google Scholar 

  22. Zhang S, Yu Y, Wang H (2015) Mittag-Leffler stability of fractional-order Hopfield neural networks. Nonlinear Anal 16:104–121

    MathSciNet  MATH  Google Scholar 

  23. Krstic M, Kokotovic PV, Kanellakopoulos I (1995) Nonlinear and adaptive control design. Wiley, New York

    MATH  Google Scholar 

Download references

Acknowledgments

Supported by the National Nature Science Foundation of China (No. 11371049) and the Fundamental Research Funds for the Central Universities (No. 2016JBM070).

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Authors and Affiliations

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

    Shuo Zhang, Yongguang Yu & Lingling Geng

Authors
  1. Shuo Zhang
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  2. Yongguang Yu
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  3. Lingling Geng
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Correspondence to Yongguang Yu.

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Zhang, S., Yu, Y. & Geng, L. Stability Analysis of Fractional-Order Hopfield Neural Networks with Time-Varying External Inputs. Neural Process Lett 45, 223–241 (2017). https://doi.org/10.1007/s11063-016-9522-1

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  • DOI: https://doi.org/10.1007/s11063-016-9522-1

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