Skip to main content
SpringerLink
Log in

On Finite-Time Stability for Fractional-Order Neural Networks with Proportional Delays

  • Published:
Neural Processing Letters Aims and scope Submit manuscript

Abstract

This paper is concerned with fractional-order neural networks with proportional delays. Applying inequality technique, some sufficient criteria which ensure the stability of such fractional-order neural networks with proportional delays over a finite-time interval are established. Computer simulations are carried out to illustrate our theoretical predictions. The derived results of this paper are new and complement some earlier ones.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Ban JC, Chang CH (2016) When are two multi-layer cellular neural networks the same? Neural Netw 79:12–19

    Article  MATH  Google Scholar 

  2. Wang LX, Zhang JM, Shao HJ (2014) Existence and global stability of a periodic solution for a cellular neural network. Commun Nonlinear Sci Numer Simul 19(9):2983–2992

    Article  MathSciNet  Google Scholar 

  3. Huang CX, Cao J, Cao JD (2016) Stability analysis of switched cellular neural networks: a mode-dependent average dwell time approach. Neural Netw 82:84–99

    Article  Google Scholar 

  4. Song QK, Yan H, Zhao ZJ, Liu YR (2016) Global exponential stability of complex-valued neural networks with both time-varying delays and impulsive effects. Neural Netw 79:108–116

    Article  MATH  Google Scholar 

  5. Yu YH (2016) Global exponential convergence for a class of HCNNs with neutral time-proportional delays. Appl Math Comput 285:1–7

    MathSciNet  MATH  Google Scholar 

  6. Yao LG (2017) Global exponential convergence of neutral type shunting inhibitory cellular neural networks with D operator. Neural Process Lett 45:401–409

  7. Xu CJ, Wu YS (2016) On almost automorphic solutions for cellular neural networks with time-varying delays in leakage terms on time scales. J Intell Fuzzy Syst 30:423–436

    Article  MATH  Google Scholar 

  8. Xu CJ, Zhang QM, Wu YS (2016) Existence and exponential stability of periodic solution to fuzzy cellular neural networks with distributed delays. Int J Fuzzy Syst 18(1):41–51

    Article  MathSciNet  Google Scholar 

  9. Xu CJ, Li PL, Pang YC (2016) Exponential stability of almost periodic solutions for memristor-based neural networks with distributed leakage delays. Neural Comput 28:2726–2756

  10. Xu CJ (2016) Existence and exponential stability of anti-periodic solution in cellular neural networks with time-varying delays and impulsive effects. Electron J Differ Equ 2016(2):1–14

    MathSciNet  Google Scholar 

  11. Balasubramaniam P, Ali MS, Arik S (2010) Global asymptotic stability of stochastic fuzzy cellular neural networks with multiple time-varying delays. Expert Syst Appl 37(12):7737–7744

    Article  Google Scholar 

  12. Balasubramaniam P, Kalpana M, Rakkiyappan R (2011) Existence and global asymptotic stability of fuzzy cellular neural networks with time dealy in the leakage term and unbounded distributed delays. Circuits Syst Signal Process 30(6):1595–1616

    Article  MATH  Google Scholar 

  13. Yang WG (2014) Periodic solution for fuzzy Cohen-Grossberg BAM neural networks with both time-varying and distributed delays and variable coefficients. Neural Process Lett 40(1):51–73

    Article  Google Scholar 

  14. Xu CJ, Li PL (2016) Existence and exponentially stability of anti-periodic solutions for neutral BAM neural networks with time-varying delays in the leakage terms. J Nonliner Sci Appl 9(3):1285–1305

    Article  MathSciNet  MATH  Google Scholar 

  15. Xu CJ, Zhang QM, Wu YS (2014) Existence and stability of pseudo almost periodic solutions for shunting inhibitory cellular neural networks with neutral type delays and time-varying leakage delays. Netw Comput Neural Syst 25(4):168–192

    Article  Google Scholar 

  16. Song QK, Zhao ZJ (2016) Stability criterion of complex-valued neural networks with both leakage delay and time-varying delays on time scales. Neurocomputing 171:179–184

    Article  Google Scholar 

  17. Stamova IM, Ilarionov R (2010) On global exponential stability for impulsive cellular neural networks with time-varying delays. Comput Math Appl 59(11):3508–3515

    Article  MathSciNet  MATH  Google Scholar 

  18. Tyagi S, Abbas S, Pinto M, Sepúlveda D (2016) WITHDRAWN: Uniform Euler approximation of solutions of fractional-order delayed cellular neural network on bounded intervals. Comput Math Appl. https://doi.org/10.1016/j.camwa.2016.04.007

  19. Abdurahman A, Jiang HJ, Teng ZD (2016) Finite-time synchronization for fuzzy cellular neural networks with time-varying delays. Fuzzy Sets Syst 297:96–111

    Article  MathSciNet  MATH  Google Scholar 

  20. Wang P, Li B, Li YK (2015) Square-mean almost periodic solutions for impulsive stochastic shunting inhibitory cellular neural networks with delays. Neurocomputing 167:76–82

    Article  Google Scholar 

  21. Rakkiyappan R, Sakthivel N, Park JH, Kwon OM (2013) Sampled-data state estimation for Markovian jumping fuzzy cellular neural networks with mode-dependent probabilistic time-varying delays. Appl Math Comput 221:741–769

    MathSciNet  MATH  Google Scholar 

  22. Balasubramaniam P, Kalpana M, Rakkiyappan R (2011) Existence and global asymptotic stability of fuzzy cellular neural networks with time delay in the leakage term and unbounded distributed delays. Circuits Syst Signal Process 30(6):1595–1616

    Article  MathSciNet  MATH  Google Scholar 

  23. Wan Y, Cao JD, Wen GH, Yu WW (2016) Robust fixed-time synchronization of delayed Cohen–Grossberg neural networks. Neural Netw 73:86–94

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  26. Bao HB, Cao JD (2016) Finite-time generalized synchronization of nonidentical delayed chaotic systems. Nonlinear Anal Model Control 21(3):306–324

    Article  MathSciNet  MATH  Google Scholar 

  27. Liu Y, Zhang DD, Lou JG, Lu JQ, Cao JD (2018) Stability analysis of quaternion-valued neural networks: Decomposition and direct approaches. IEEE Trans Neural Netw Learn Syst. https://doi.org/10.1109/TNNLS.2017.2755697 (in press)

  28. Liu Y, Zhang DD, Lu JQ, Cao JD (2016) Global \(\mu \)-stability criteria for quaternion-valued neural networks with unbounded time-varying delays. Inf Sci 360:273–288

    Article  Google Scholar 

  29. Liu Y, Zhang DD, Lu JQ (2017) Global exponential stability for quaternion-valued recurrent neural networks with time-varying delays. Nonlinear Dyn 87(1):553–565

    Article  MATH  Google Scholar 

  30. Liu Y, Xu P, Lu JQ, Liang JL (2016) Global stability of Clifford-valued recurrent neural networks with time delays. Nonlinear Dyn 84(2):767–777

    Article  MathSciNet  MATH  Google Scholar 

  31. Yang RJ, Wu B, Liu Y (2015) A Halanay-type inequality approach to the stability analysis of discrete-time neural networks with delays. Appl Math Comput 265:696–707

    MathSciNet  MATH  Google Scholar 

  32. Tao W, Liu Y, Lu JQ (2017) Stability and \(L_2\)-gain analysis for switched singular linear systems with jumps. Math Methods Appl Sci 40(3):589–599

    Article  MathSciNet  MATH  Google Scholar 

  33. Li XD, Song SJ (2013) Impulsive control for existence, uniqueness and global stability of periodic solutions of recurrent neural networks with discrete and continuously distributed delays. IEEE Trans Neural Netw Learn Syst 24:868–877

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, New York

    MATH  Google Scholar 

  36. Sabatier J, Agrawal OP, Machado J (2007) Theoretical development and applications. Advance in fractional calculus. Springer, Berlin

    Google Scholar 

  37. Podlubny I (1999) Fractional differential equations. Academic Press, San Diego

    MATH  Google Scholar 

  38. Buter PL, Westphal U (2000) An introduction to fractional calculus. World Scientific, Singapore

    Google Scholar 

  39. Lundstrom BN, Higgs MH, Spain WJ, Fairhall AL (2008) Fractional differentiation by neocortical pyramidal neurons. Nat Neurosci 11(11):1335–1342

    Article  Google Scholar 

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

    Article  Google Scholar 

  41. Anastassiou GA (2012) Fractional neural network approximation. Comput Math Appl 64(6):1655–1676

    Article  MathSciNet  MATH  Google Scholar 

  42. Wu AL, Zeng ZG (2016) Boundedness, Mittag–Leffler stability and asymptotical \(\alpha \)-periodicity of fractional-order fuzzy neural networks. Neural Netw 74:73–84

    Article  MATH  Google Scholar 

  43. Zhang S, Yu YG, Wang Q (2016) Stability analysis of fractional-order Hopfield neural networks with discontinuous activation functions. Neurocomputing 171:1075–1084

    Article  Google Scholar 

  44. Chen LP, Wu RC, Cao JD, Liu JB (2015) Stability and synchronization of memristor-based fractional-order delayed neural networks. Neural Netw 71:37–44

    Article  MATH  Google Scholar 

  45. Wang F, Yang YQ, Hu MF (2015) Asymptotic stability of delayed fractional-order neural networks with impulsive effects. Neurocomputing 154:239–244

    Article  Google Scholar 

  46. Huang X, Zhao Z, Wang Z, Lia YX (2012) Chaos and hyperchaos in fractional-order cellular neural networks. Neurocomputing 94:13–21

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  48. Chen L, Chai Y, Wu R, Ma T, Zhai H (2013) Dynamic analysis of a class of fractional-order neural networks with delay. Neurocomputing 111:190–194

    Article  Google Scholar 

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

    Article  Google Scholar 

  50. Li MM, Wang JR (2018) Exploring delayed Mittag–Leffler type matrix function to study finite time stability of fractional delay differential equations. Appl Math Comput 324:254–265

    MathSciNet  MATH  Google Scholar 

  51. Zhang XG, Liu LS, Wu YH (2012) Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives. Appl Math Comput 219(4):1420–1433

    MathSciNet  MATH  Google Scholar 

  52. Zhang XG, Liu LS, Wu YH (2012) The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives. Appl Math Comput 218(17):8526–8536

    MathSciNet  MATH  Google Scholar 

  53. Zhang LH, Zheng ZW (2017) Lyapunov type inequalities for the Riemann–Liouville fractional differential equations of higher order. Adv Differ Equ 2017:270. https://doi.org/10.1186/s13662-017-1329-5

    Article  MathSciNet  MATH  Google Scholar 

  54. Feng QH, Meng FW (2016) Explicit solutions for space-time fractional partial differential equations in mathematical physics by a new generalized fractional Jacobi elliptic equation-based sub-equation method. Optik 127:7450–7458

    Article  Google Scholar 

  55. Li MM, Wang JR (2017) Finite time stability of fractional delay differential equations. Appl Math Lett 64:170–176

    Article  MathSciNet  MATH  Google Scholar 

  56. Hei XD, Wu RC (2016) Finite-time stability of impulsive fractional-order systems with time-delay. Appl Math Model 40(7C8):4285–4290

    Article  MathSciNet  Google Scholar 

  57. Ma YJ, Wu BW, Wang YE (2016) Finite-time stability and finite-time boundedness of fractional order linear systems. Neurocomputing 173(3):2076–2082

    Article  Google Scholar 

  58. Yang XJ, Song QK, Liu YR, Zhao ZJ (2015) Finite-time stability analysis of fractional-order neural networks with delay. Neurocomputing 152:19–26

    Article  Google Scholar 

  59. Efimov D, Polyakov A, Fridman E, Perruquetti W, Richard JP (2014) Comments on finite-time stability of time-delay systems. Automatica 50(7):1944–1947

    Article  MathSciNet  MATH  Google Scholar 

  60. Chen LP, Liu C, Wu RC, He YG, Chai Y (2016) Finite-time stability criteria for a class of fractional-order neural networks with delay. Neural Comput Appl 27:549–556

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

    Article  Google Scholar 

  62. Zhou LQ (2015) Novel global exponential stability criteria for hybrid BAM neural networks with proportional delays. Neurocomputing 161:99–106

    Article  Google Scholar 

  63. Hien LV, Son DT (2015) Finite-time stability of a class of non-autonomous neural networks with heterogeneous proportional delays. Appl Math Comput 251:14–23

    MathSciNet  MATH  Google Scholar 

  64. Zhou LQ, Zhang YY (2016) Global exponential stability of a class of impulsive recurrent neural networks with proportional delays via fixed point theory. J Frankl Inst 353(2):561–575

    Article  MathSciNet  MATH  Google Scholar 

  65. Ockendon JR, Tayler AB (1971) The dynamics of a current collection system for an electric locomotive. Proc R Soc Lond Ser A Math Phys Sci 322(1551):447–468

    Article  Google Scholar 

  66. Song XL, Zhao P, Xing ZW, Peng JG (2016) Global asymptotic stability of CNNs with impulses and multi-proportional delays. Math Methods Appl Sci 39(4):722–733

    Article  MathSciNet  MATH  Google Scholar 

  67. Derfel GA (1982) On the behaviour of the solutions of functional and functional-differential equations with several deviating arguments. Ukr Math J 34:286–291

    Article  Google Scholar 

  68. Fox L, Ockendon DF, Tayler AB (1971) On a functional-differential equations. J Inst Math Appl 8(3):271–307

    Article  MathSciNet  MATH  Google Scholar 

  69. Derfel GA (1990) Kato problem for functional-differential equations and difference Schrodinger operator. Oper Theory 46:319–321

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  71. Kuczma M (2009) An introduction to the theory of functional equations and inequalities: Cauchy’s equation and Jensen’s inequalities. Birkhauser, Basel

    Book  MATH  Google Scholar 

  72. Corduneanu C (1971) Principles of differential and integral equations. Allyn and Bacon, Boston, MA

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  74. Li YC, Yang XS, Shi L (2016) Finite-time synchronization for competitive neural networks with mixed delays and nonidentical perturbations. Neurocomputing 85:242–253

    Article  Google Scholar 

  75. Yang XS, Ho DWC (2016) Synchronization of delayed memristive neural networks: robust analysis approach. IEEE Trans Cybern 46(12):3377–3387

    Article  Google Scholar 

  76. Shi L, Yang XS, Li YC, Feng ZZ (2016) Finite-time synchronization of nonidentical chaotic systems with multiple time-varying delays and bounded perturbations. Nonlinear Dyn 83(1–2):75–87

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang, 550004, People’s Republic of China

    Changjin Xu

  2. School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471023, People’s Republic of China

    Peiluan Li

Authors
  1. Changjin Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Peiluan Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Changjin Xu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported by National Natural Science Foundation of China (No. 61673008) and Project of High-level Innovative Talents of Guizhou Province ([2016]5651) and Major Research Project of The Innovation Group of The Education Department of Guizhou Province ([2017]039), Project of Key Laboratory of Guizhou Province with Financial and Physical Features ([2017]004) and the Foundation of Science and Technology of Guizhou Province ([2018]1025 and [2018]1020).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, C., Li, P. On Finite-Time Stability for Fractional-Order Neural Networks with Proportional Delays. Neural Process Lett 50, 1241–1256 (2019). https://doi.org/10.1007/s11063-018-9917-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11063-018-9917-2

Keywords

Search

Navigation