Skip to main content
Log in

Global Mittag-Leffler stability and synchronization of impulsive fractional-order neural networks with time-varying delays

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper we consider a class of impulsive Caputo fractional-order cellular neural networks with time-varying delays. Applying the fractional Lyapunov method and Mittag-Leffler functions, we give sufficient conditions for global Mittag-Leffler stability which implies global asymptotic stability of the network equilibrium. Our results provide a design method of impulsive control law which globally asymptotically stabilizes the impulse free fractional-order neural network time-delay model. The synchronization of fractional chaotic networks via non-impulsive linear controller is also considered. Illustrative examples are given to demonstrate the effectiveness of the obtained results.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arbib, M.: Branins, Machines, and Mathematics. Springer, New York (1987)

    Book  Google Scholar 

  2. Haykin, S.: Neural Networks: A Comprehensive Foundation. Prentice-Hall, Englewood Cliffs, New Jersey (1998)

    Google Scholar 

  3. Chua, L.O., Yang, L.: Cellular neural networks: theory. IEEE Trans. Circuits Syst. 35, 1257–1272 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chua, L.O., Yang, L.: Cellular neural networks: applications. IEEE Trans. Circuits Syst. 35, 1273–1290 (1988)

    Article  MathSciNet  Google Scholar 

  5. Arik, S., Tavsanoglu, V.: On the global asymptotic stability of delayed cellular neural networks. IEEE Trans. Circuits Syst. I(47), 571–574 (2000)

    Article  MathSciNet  Google Scholar 

  6. Wang, L., Cao, J.: Global robust point dissipativity of interval neural networks with mixed time-varying delays. Nonlinear Dyn. 55, 169–178 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Zhang, Q., Wei, X., Xu, J.: On global exponential stability of delayed cellular neural networks with time-varying delays. Appl. Math. Comput. 162, 679–686 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Long, S., Xu, D.: Delay-dependent stability analysis for impulsive neural networks with time varying delays. Neurocomputing 71, 1705–1713 (2008)

    Article  Google Scholar 

  9. Stamov, G.T.: Impulsive cellular neural networks and almost periodicity. Proc. Jpn. Acad. Ser. A Math. Sci. 80, 198–203 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Stamov, G.T.: Almost Periodic Solutions of Impulsive Differential Equations. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

  11. Stamov, G.T., Stamova, I.M.: Almost periodic solutions for impulsive neural networks with delay. Appl. Math. Model. 31, 1263–1270 (2007)

    Article  MATH  Google Scholar 

  12. Stamova, I.M.: Stability Analysis of Impulsive Functional Differential Equations. Walter de Gruyter, Berlin (2009)

    Book  MATH  Google Scholar 

  13. Wang, Q., Liu, X.: Exponential stability of impulsive cellular neural networks with time delay via Lyapunov functionals. Appl. Math. Comput. 194, 186–198 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Wang, X., Li, S., Xu, D.: Globally exponential stability of periodic solutions for impulsive neutral-type neural networks with delays. Nonlinear Dyn. 64, 65–75 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. Khadra, A., Liu, X., Shen, X.: Impulsive control and synchronization of spatiotemporal chaos. Chaos Solitons Fractals 26, 615–636 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Litak, G., Ali, M., Saha, L.M.: Pulsating feedback control for stabilizing unstable periodic orbits in a nonlinear oscillator with a non-symmetric potential. Int. J. Bifurcation Chaos 17, 2797–2803 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  17. Litak, G., Borowiec, M., Ali, M., Saha, L.M., Friswell, M.I.: Pulsive feedback control of a quarter car model forced by a road profile. Chaos Solitons Fractals 33, 1672–1676 (2007)

    Article  Google Scholar 

  18. Stamova, I.M., Stamov, G.T.: Impulsive control on global asymptotic stability for a class of bidirectional associative memory neural networks with distributed delays. Math. Comput. Model. 53, 824–831 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  19. Stamova, I.M., Stamov, T., Simeonova, N.: Impulsive control on global exponential stability for cellular neural networks with supremums. J. Vib. Control 19, 483–490 (2013)

    Article  MathSciNet  Google Scholar 

  20. Sun, J., Han, Q.L., Jiang, X.: Impulsive control of time-delay systems using delayed impulse and its application to impulsive masterslave synchronization. Phys. Lett. A 372, 6375–6380 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. Diethelm, K.: The Analysis of Fractional Differential Equations. An Application-oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin (2010)

    MATH  Google Scholar 

  22. Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. Elsevier, New York (2006)

    MATH  Google Scholar 

  23. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  24. Babakhani, A., Baleanu, D., Khanbabaie, R.: Hopf bifurcation for a class of fractional differential equations with delay. Nonlinear Dyn. 69, 721–729 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  25. Benchohra, M., Henderson, J., Ntouyas, S.K., Ouahab, A.: Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl. 338, 1340–1350 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  26. Bhalekar, S., Daftardar-Gejji, V., Baleanu, D., Magin, R.: Generalized fractional order bloch equation with extended delay. Int. J. Bifurcation Chaos 22, 1250071 (2012)

    Article  Google Scholar 

  27. Abbas, S., Benchohra, M.: Impulsive partial hyperbolic functional differential equations of fractional order with state-dependent delay. Fract. Calc. Appl. Anal. 13, 225–244 (2010)

    MATH  MathSciNet  Google Scholar 

  28. Chen, F., Chen, A., Wang, X.: On the solutions for impulsive fractional functional differential equations. Differ. Equ. Dyn. Syst. 17, 379–391 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  29. Wang, H.: Existence results for fractional functional differential equations with impulses. J. Appl. Math. Comput. 38, 85–101 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  30. Lu, J.G., Chen, Y.Q.: Stability and stabilization of fractional-order linear systems with convex polytopic uncertainties. Fract. Calc. Appl. Anal. 16, 142–157 (2013)

    Article  MathSciNet  Google Scholar 

  31. Stamova, I., Stamov, G.: Lipschitz stability criteria for functional differential systems of fractional order. J. Math. Phys. 54, 043502 (2013)

    Article  MathSciNet  Google Scholar 

  32. Stamova, I.M., Stamov, G.T.: Stability analysis of impulsive functional systems of fractional order. Commun. Nonlinear Sci. Numer. Simulat. 19, 702–709 (2014)

    Article  MathSciNet  Google Scholar 

  33. Zeng, C., Chen, Y.Q., Yang, Q.: Almost sure and moment stability properties of fractional order Black–Scholes model. Fract. Calc. Appl. Anal. 16, 317–331 (2013)

    Article  MathSciNet  Google Scholar 

  34. Li, C., Deng, W., Xu, D.: Chaos synchronization of the Chua system with a fractional order. Physica A 360, 171–185 (2006)

    Article  MathSciNet  Google Scholar 

  35. Razminia, A., Baleanu, D.: Fractional synchronization of chaotic systems with different orders. Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci. 13, 314–321 (2012)

    MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  37. Chen, L., Qu, J., Chai, Y., Wu, R., Qi, G.: Synchronization of a class of fractional-order chaotic neural networks. Entropy 15, 3265–3276 (2013)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  40. Wu, X., Lai, D., Lu, H.: Generalized synchronization of the fractional-order chaos in weighted complex dynamical networks with nonidentical nodes. Nonlinear Dyn. 69, 667–683 (2012)

    Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  44. Wu, R., Hei, X., Chen, L.: Finite-time stability of fractional-order neural networks with delay. Commun. Theor. Phys. 60, 189–193 (2013)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  46. Razumikhin, B.S.: Stability of Hereditary Systems. Nauka, Moscow (1988). (in Russian)

    Google Scholar 

  47. Yan, J., Shen, J.: Impulsive stabilization of impulsive functional differential equations by Lyapunov-Razumikhin functions. Nonlinear Anal. 37, 245–255 (1999)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ivanka Stamova.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Stamova, I. Global Mittag-Leffler stability and synchronization of impulsive fractional-order neural networks with time-varying delays. Nonlinear Dyn 77, 1251–1260 (2014). https://doi.org/10.1007/s11071-014-1375-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-014-1375-4

Keywords

Navigation