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Finite-time Stability of Fractional-order Complex-valued Neural Networks with Time Delays

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Abstract

The analysis of finite-time stability for a class of fractional-order complex valued neural networks with delays is considered in this paper. Utilizing Gronwall inequality, Cauchy-Schiwartz inequality and inequality scaling techniques, some sufficient conditions for guaranteeing the finite-time stability of the system are derived respectively under two cases with order \(1/2\le \alpha < 1\) and \(0<\alpha <1/2\), in which different inequality scaling strategies are employed. Two numerical examples are also proposed to demonstrate the validity and feasibility of the obtained results.

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References

  1. Leibniz GW (1962) Mathematische schiften. Georg Olms Verlagsbuch-handlung, Hildesheim

    Google Scholar 

  2. Hall MG, Barrick TR (2008) From diffusion-weighted MRI to anomalous diffusion imaging. Magn Resonan Med 59(3):55–447

    Article  Google Scholar 

  3. Enacheanu O, Riu D, Retiere N, et al (2006) Identification of fractional order models for electrical networks. In: IECON 2006-32nd annual conference on IEEE industrial electronics, pp. 5392–5396

  4. Ahn HS, Chen YQ (2008) Necessary and sufficient stability condition of fractional-order interval linear systems. Automatica 44(11):2985–2988

    Article  MathSciNet  MATH  Google Scholar 

  5. Rostek S, Schöbel R (2013) A note on the use of fractional Brownian motion for financial modeling. Econ Model 30:30–35

    Article  Google Scholar 

  6. Anastasio TJ (1994) The fractional-order dynamics of brainstem vestibulo-oculomotor neurons. Biol Cybern 72(1):69–79

    Article  Google Scholar 

  7. Arena P, Caponetto R, Fortuna L, Porto D (1998) Bifurcation and chaos in noninteger order cellular neural networks. Int J Bifurc Chaos 8(7):1527–1539

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  9. Arena P, Fortuna L, Porto D (2000) Chaotic behavior in noninteger order cellular neural networks. Physi Rev E 61(1):776–781

    Article  Google Scholar 

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

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

    Article  MATH  Google Scholar 

  13. Petráš I (2006) A note on the fractional-order cellular neural networks. In: The 2006 international joint conference on neural networks (IJCNN), IEEE, pp. 1021–1024

  14. Hirose A (2013) Complex-valued neural networks: advances and applications. Wiley, Hoboken

    Book  Google Scholar 

  15. Bao HB, Park JH, Cao JD (2016) Synchronization of fractional-order complex-valued neural networks with time delay. Neural Netw 81:16–28

    Article  Google Scholar 

  16. Song QK, Zhao ZJ, Liu YR (2015) Impulsive effects on stability of discrete-time complex-valued neural networks with both discrete and distributed time-varying delays. Neurocomputing 168:1044–1050

    Article  Google Scholar 

  17. Jankowski S, Lozowski A, Zurada JM (1996) Complex-valued multistate neural associative memory. IEEE Trans Neural Netw 7(6):1491–1496

    Article  Google Scholar 

  18. Mathews JH, Howell RW (2012) Complex analysis for mathematics and engineering, 6th edn. Jones and Bartlett Learning, Burlington

    MATH  Google Scholar 

  19. Xu XH, Zhang JY, Shi JZ (2014) Exponential stability of complex-valued neural networks with mixed delays. Neurocomputing 128:483–490

    Article  Google Scholar 

  20. Gong WQ, Liang JL, Zhang CJ (2016) Multistability of complex-valued neural networks with distributed delays. Neural Computing and Applications. doi:10.1007/s00521-016-2305-9

  21. Zhou B, Song QK (2013) Boundedness and complete stability of complex-valued neural networks with time delay. IEEE Trans Neural Netw Learn Syst 24(8):1227–1238

    Article  Google Scholar 

  22. Wu EL, Yang XS (2016) Adaptive synchronization of coupled nonidentical chaotic systems with complex variables and stochastic perturbations. Nonlinear Dyn 84(1):261–269

    Article  MathSciNet  MATH  Google Scholar 

  23. Rakkiyappana R, Velmurugana G, Cao JD (2015) Stability analysis of fractional-order complex-valued neural networks with time delays. Chaos Solitons Fractals 78:297–316

    Article  MathSciNet  Google Scholar 

  24. Rakkiyappan R, Cao JD, Velmurugan G (2015) Existence and uniform stability analysis of fractional-order complex-valued neural networks with time delays. IEEE Trans Neural Netw Learn Syst 26(1):84–97

    Article  MathSciNet  MATH  Google Scholar 

  25. Weiss L, Infante F (1965) On the stability of systems defined over finite time interval. Proc Natl Acad Sci 54(1):44–48

    Article  MathSciNet  MATH  Google Scholar 

  26. Khoo S, Yin JL, Man ZH, Yu XH (2013) Finite-time stabilization of stochastic nonlinear systems instrict-feedback form. Automatica 49:1403–1410

    Article  MathSciNet  MATH  Google Scholar 

  27. Liu H, Zhao XD, Zhang HM (2014) New approaches to finite-time stability and stabilization for nonlinear system. Neurocomputing 138:218–228

    Article  Google Scholar 

  28. Hien LV (2014) An explicit criterion for finite-time stability of linear nonautonomous systems with delays. Appl Math Lett 30:12–18

    Article  MathSciNet  MATH  Google Scholar 

  29. Yang XS, Ho DWC, Lu JQ, Song Q (2015) Finite-time cluster synchronization of T-S fuzzy complex networks with discontinuous subsystems and random coupling delays. IEEE Trans Fuzzy Syst 23(6):2302–2316

    Article  Google Scholar 

  30. Yang XS, Lu JQ (2016) Finite-time synchronization of coupled networks with Markovian topology and impulsive effects. IEEE Trans Autom Control 61(8):2256–2261

    Article  MathSciNet  MATH  Google Scholar 

  31. Wu RC, Hei XD, Chen LP (2013) Finite-time stability of fractional-order neural networks with delay. Commun Theor Phys 60(2):189–193

    Article  MathSciNet  MATH  Google Scholar 

  32. Wu RC, Lu YF, Chen LP (2015) Finite-time stability of fractional delayed neural networks. Neurocomputing 149:700–707

    Article  Google Scholar 

  33. Lazarević MP (2006) Finite time stability analysis of PD\(^\alpha \) fractional control of robotic time-delay systems. Mech Res Commun 33:269–279

    Article  MathSciNet  MATH  Google Scholar 

  34. Lazarević MP, Spasić AM (2009) Finite-time stability analysis of fractional order time-delay systems: Gronwall’s approach. Math Comput Model 49:475–481

    Article  MathSciNet  MATH  Google Scholar 

  35. Diethelm K (2004) The analysis of fractional differential equations: an application-oriented exposition using differential operators of caputo type. Springer-Verlag, New York

    MATH  Google Scholar 

  36. Li CP, Deng WH (2007) Remarks on fractional derivatives. Appl Math Comput 187(2):777–784

    MathSciNet  MATH  Google Scholar 

  37. Jiang ZJ, Wu ZQ (1994) The theory of real variable functions. Higher Education Press, Beijing

    Google Scholar 

  38. Kuczma M (2009) An introduction to the theory of functional equations and inequalities: Cauchy’s equation and Jensen’s inequality, Birkhäuser

  39. Corduneanu C (1977) Principles of differential and intergral equations, 2nd edn. Chelsea Pub Co, New York

    Google Scholar 

  40. La Salle S (1961) Lefschet, stability by Lyapunovs direct method. Academic Press, New York

    Google Scholar 

  41. Lam L, Weiss L (1972) Finite time stability with respect to time-varying sets. J Frankl Inst 9:415–421

    MathSciNet  MATH  Google Scholar 

  42. Grujić Lj T (1975) Non-Lyapunov stability analysis of large-scale systems on time-varying sets. Int J Control 21(3):401–415

    Article  MathSciNet  MATH  Google Scholar 

  43. Grujić Lj T (1975) Practical stability with settling time on composite systems. Automatika(Yu) 9:1–11

    Google Scholar 

  44. Bhat SP, Bernstein DS (1995) Lyapunov analysis of finite-time differential equations. In: Proceedings of the American control conference, Seattle, pp. 1831–1832

  45. Cao JD, Li RX (2017) Fixed-time synchronization of delayed memristor-based recurrent neural networks. Sci China Inf Sci 60(3):032201. doi:10.1007/s11432-016-0555-2

    Article  Google Scholar 

  46. Diethelm K, Ford NJ, Freed AD (2002) Apredictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn 29(1):3–22

    Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Mathematics, and Research Center for Complex Systems and Network Sciences, Southeast University, Nanjing, 210096, China

    Xiaoshuai Ding, Jinde Cao & Xuan Zhao

  2. School of Education, Xizang Minzu University, Xianyang, 712082, China

    Xiaoshuai Ding

  3. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Jinde Cao

  4. Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Fuad E. Alsaadi

Authors
  1. Xiaoshuai Ding
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  2. Jinde Cao
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  3. Xuan Zhao
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  4. Fuad E. Alsaadi
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Corresponding author

Correspondence to Jinde Cao.

Additional information

This work was jointly supported by National Natural Science Foundation of China (Nos. 11301308, 61573096, 61272530), the 333 Engineering Foundation of Jiangsu Province of China (No. BRA2015286), and the Natural Science Youth Foundation of Jiangsu Province of China (No. BK20160660).

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Ding, X., Cao, J., Zhao, X. et al. Finite-time Stability of Fractional-order Complex-valued Neural Networks with Time Delays. Neural Process Lett 46, 561–580 (2017). https://doi.org/10.1007/s11063-017-9604-8

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  • DOI: https://doi.org/10.1007/s11063-017-9604-8

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