Advertisement

Cognitive Neurodynamics

, Volume 9, Issue 2, pp 145–177 | Cite as

Stability analysis of memristor-based fractional-order neural networks with different memductance functions

  • R. Rakkiyappan
  • G. Velmurugan
  • Jinde Cao
Research Article

Abstract

In this paper, the problem of the existence, uniqueness and uniform stability of memristor-based fractional-order neural networks (MFNNs) with two different types of memductance functions is extensively investigated. Moreover, we formulate the complex-valued memristor-based fractional-order neural networks (CVMFNNs) with two different types of memductance functions and analyze the existence, uniqueness and uniform stability of such networks. By using Banach contraction principle and analysis technique, some sufficient conditions are obtained to ensure the existence, uniqueness and uniform stability of the considered MFNNs and CVMFNNs with two different types of memductance functions. The analysis results establish from the theory of fractional-order differential equations with discontinuous right-hand sides. Finally, four numerical examples are presented to show the effectiveness of our theoretical results.

Keywords

Fractional-order Memristor-based neural networks Banach contraction principle Time delays 

Notes

Acknowledgments

This work was supported by NBHM research Project No. 2/48(7)/2012/NBHM(R.P.)/R and D-II/12669, the National Natural Science Foundation of China under Grant Nos. 61272530 and 11072059, the Natural Science Foundation of Jiangsu Province of China under Grant No. BK2012741, and the Specialized Research Fund for the Doctoral Program of Higher Education under Grant Nos. 20110092110017 and 20130092110017.

References

  1. Ahmeda E, Elgazzar AS (2007) On fractional order differential equations model for nonlocal epidemics. Phys A 379:607–614CrossRefGoogle Scholar
  2. Boroomand A, Menhaj M (2009) Fractional-order Hopfield neural networks. Lecture notes in computer science vol 5506, pp 883–890Google Scholar
  3. Bouzerdoum A, Pattison TR (1993) Neural network for quadratic optimization with bound constraints. IEEE Trans Neural Netw 4:293–303CrossRefPubMedGoogle Scholar
  4. Cai Z, Huang L (2014) Functional differential inclusions and dynamic behaviors for memristor-based BAM neural networks with time-varying delays. Commun Nonlinear Sci Numer Simul 19:1279–1300CrossRefGoogle Scholar
  5. Chen X, Song Q (2013) Global stability of complex-valued neural networks with both leakage time delay and discrete time delay on time scales. Neurocomputing 121:254–264CrossRefGoogle Scholar
  6. Chen L, Chai Y, Wu R, Ma T, Zhai H (2013) Dynamics analysis of a class of fractional-order neural networks with delay. Neurocomputing 111:190–194CrossRefGoogle Scholar
  7. Chen J, Zeng Z, Jiang P (2014) Global Mittag–Leffler stability and synchronization of memristor-based fractional-order neural networks. Neural Netw 51:1–8CrossRefPubMedGoogle Scholar
  8. Chua LO (1971) Memristor-the missing circuit element. IEEE Trans Circuit Theory 18:507–519CrossRefGoogle Scholar
  9. Delavari H, Baleanu D, Sadati J (2012) Stability analysis of Caputo fractional-order nonlinear systems revisited. Nonlinear Dyn 67:2433–2439CrossRefGoogle Scholar
  10. Deng WH, Li CP (2005) Chaos synchronization of the fractional Lu system. Phys A 353:61–72CrossRefGoogle Scholar
  11. Duan CJ, Song O (2010) Boundedness and stability for discrete-time delayed neural networks with complex-valued linear threshold neurons. Discrete Dyn Nat Soc Article ID:368379:1–19Google Scholar
  12. Filippov A (1988) Differential equations with discontinuous right-hand sides. Kluwer, DordrechtCrossRefGoogle Scholar
  13. Guo D, Li C (2012) Population rate coding in recurrent neuronal networks with unreliable synapses. Cogn Neurodyn 6:75–87CrossRefPubMedCentralPubMedGoogle Scholar
  14. Guo Z, Wang J, Yan Z (2013) Global exponential dissipativity and stabilization of memristor-based recurrent neural networks with time-varying delays. Neural Netw 48:158–172CrossRefPubMedGoogle Scholar
  15. Hilfer R (2000) Applications of fractional calculus in physics. World Scientific York, SingaporeCrossRefGoogle Scholar
  16. Hirose A (2012) Complex-valued neural networks. Springer, BerlinCrossRefGoogle Scholar
  17. Hu J, Wang J (2012) Global stability of complex-valued recurrent neural networks with time-delays. IEEE Trans Neural Netw Learn Syst 23:853–865CrossRefPubMedGoogle Scholar
  18. Huang Y, Zhang H, Wang Z (2014) Multistability of complex-valued recurrent neural networks with real-imaginary-type activation functions. Appl Math Comput 229:187–200CrossRefGoogle Scholar
  19. Kaslik E, Sivasundaram S (2012) Nonlinear dynamics and chaos in fractional-order neural networks. Neural Netw 32:245–256CrossRefPubMedGoogle Scholar
  20. Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and application of fractional differential equations. Elsevier, New YorkGoogle Scholar
  21. Kosko B (1988) Bidirectional associative memories. IEEE Trans Syst Man Cybern 18:49–60CrossRefGoogle Scholar
  22. Laskin N (2000) Fractional market dynamics. Phys A 287:482–492CrossRefGoogle Scholar
  23. Mathews JH, Howell RW (1997) Complex analysis for mathematics and engineering, 3rd edn. Jones and Bartlett Publ Inc, Boston, MAGoogle Scholar
  24. Nitta T (2004) Orthogonality of decision boundaries of complex-valued neural networks. Neural Comput 16:73–97CrossRefPubMedGoogle Scholar
  25. Peng GJ, Jiang YL, Chen F (2008) Generalized projective synchronization of fractional order chaotic systems. Phys A 387:3738–3746CrossRefGoogle Scholar
  26. Podlubny I (1999) Fractional differential equations. Academic Press, San DiegoGoogle Scholar
  27. Qi J, Li C, Huang T (2014) Stability of delayed memristive neural networks with time-varying impulses. Cogn Neurodyn 8:429–436Google Scholar
  28. Rao VSH, Murthy GR (2009) Global dynamics of a class of complex valued neural networks. Int J Neural Syst 18:165–171Google Scholar
  29. Seow MJ, Asari VK, Livingston A (2010) Learning as a nonlinear line of attraction in a recurrent neural network. Neural Comput Appl 19:337–342CrossRefGoogle Scholar
  30. Strukov DB, Snider GS, Sterwart DR, Williams RS (2008) The missing memristor found. Nature 453:80–83CrossRefPubMedGoogle Scholar
  31. Tanaka G, Aihara K (2009) Complex-valued multistate associative memory with nonlinear multilevel functions for gray-level image reconstruction. IEEE Trans Neural Netw 20:1463–1473CrossRefPubMedGoogle Scholar
  32. Tour JM, He T (2008) The fourth element. Nature 453:42–43CrossRefPubMedGoogle Scholar
  33. Wen S, Zeng Z, Huang T, Chen Y (2013) Passivity analysis of memristor-based recurrent neural networks with time-varying delays. J Frankl Inst 350:2354–2370CrossRefGoogle Scholar
  34. Wu XJ, Lu HT, Shen SL (2009) Synchronization of a new fractional-order hyperchaotic system. Phys Lett A 373:2329–2337CrossRefGoogle Scholar
  35. Wu A, Zeng Z, Zhu X, Zhang J (2011) Exponential synchronization of memristor-based recurrent neural networks with time delays. Neurocomputing 74:3043–3050CrossRefGoogle Scholar
  36. Wu H, Liao X, Feng W, Guo S (2012) Mean square stability of uncertain stochastic BAM neural networks with interval time-varying delays. Cogn Neurodyn 6:443–458CrossRefPubMedCentralPubMedGoogle Scholar
  37. Wu H, Zhang L, Ding S, Guo X, Wang L (2013a) Complete periodic synchronization of memristor-based neural networks with time-varying delays. Discrete Dyn Nat Soci Article ID:140153:1–12Google Scholar
  38. Wu A, Zeng Z, Xiao J (2013b) Dynamic evolution evoked by external inputs in memristor-based wavelet neural networks with different memductance functions. Adv Differ Equ 258:1–14Google Scholar
  39. Wu A, Zeng Z (2012) Dynamic behaviors of memristor-based recurrent neural networks with time-varying delays. Neural Netw 36:1–10CrossRefPubMedGoogle Scholar
  40. Wu A, Zeng Z (2013) Anti-synchronization control of a class of memristive recurrent neural networks. Commun Nonlinear Sci Numer Simul 18:373–385CrossRefGoogle Scholar
  41. Wu A, Zeng Z (2014) Passivity analysis of memristive neural networks with different memductance functions. Commun Nonlinear Sci Numer Simul 19:274–285CrossRefGoogle Scholar
  42. Xu X, Zhang J, Shi J (2014) Exponential stability of complex-valued neural networks with mixed delays. Neurocomputing 128:483–490Google Scholar
  43. Yang X, Cao J, Ho DWC (2014) Exponential synchronization of discontinuous neural networks with time-varying mixed delays via state feedback and impulsive control. Cogn Neurodyn. doi: 10.1007/s11571-014-9307-z
  44. Yang X, Cao J, Yu W (2014) Exponential synchronization of memristive Cohen–Grossberg neural networks with mixed delays. Cogn Neurodyn 8:239–249CrossRefPubMedGoogle Scholar
  45. Yu J, Hu C, Jiang H (2012) \(\alpha \)-stability and \(\alpha \)-synchronization for fractional-order neural networks. Neural Netw 35:82–87CrossRefPubMedGoogle Scholar
  46. Zhang G, Shen Y, Wang L (2013) Global anti-synchronization of a class of chaotic memristive neural networks with time-varying delays. Neural Netw 46:1–8CrossRefPubMedGoogle Scholar
  47. Zhou B, Song Q (2013) Boundedness and complete stability of complex-valued neural networks with time delay. IEEE Trans Neural Netw Learn Syst 24:1227–1238CrossRefGoogle Scholar
  48. Zou T, Qu J, Chen L, Chai Y, Yang Z (2014) Stability analysis of a class of fractional-order neural networks. TELKOMNIKA Indones J Electr Eng 12:1086–1093Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsBharathiar UniversityCoimbatoreIndia
  2. 2.Department of Mathematics, and Research Center for Complex Systems and Network SciencesSoutheast UniversityNanjingChina
  3. 3.Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia

Personalised recommendations