Nonlinear Dynamics

, Volume 77, Issue 4, pp 1251–1260 | Cite as

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

  • Ivanka Stamova
Original Paper


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.


Global Mittag-Leffler stability  Synchronization Neural networks Fractional-order derivatives Time-varying delays Impulsive control Lyapunov method 


  1. 1.
    Arbib, M.: Branins, Machines, and Mathematics. Springer, New York (1987)CrossRefGoogle Scholar
  2. 2.
    Haykin, S.: Neural Networks: A Comprehensive Foundation. Prentice-Hall, Englewood Cliffs, New Jersey (1998)Google Scholar
  3. 3.
    Chua, L.O., Yang, L.: Cellular neural networks: theory. IEEE Trans. Circuits Syst. 35, 1257–1272 (1988)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Chua, L.O., Yang, L.: Cellular neural networks: applications. IEEE Trans. Circuits Syst. 35, 1273–1290 (1988)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Arik, S., Tavsanoglu, V.: On the global asymptotic stability of delayed cellular neural networks. IEEE Trans. Circuits Syst. I(47), 571–574 (2000)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Wang, L., Cao, J.: Global robust point dissipativity of interval neural networks with mixed time-varying delays. Nonlinear Dyn. 55, 169–178 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Long, S., Xu, D.: Delay-dependent stability analysis for impulsive neural networks with time varying delays. Neurocomputing 71, 1705–1713 (2008)CrossRefGoogle Scholar
  9. 9.
    Stamov, G.T.: Impulsive cellular neural networks and almost periodicity. Proc. Jpn. Acad. Ser. A Math. Sci. 80, 198–203 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Stamov, G.T.: Almost Periodic Solutions of Impulsive Differential Equations. Springer, Berlin (2012)zbMATHCrossRefGoogle Scholar
  11. 11.
    Stamov, G.T., Stamova, I.M.: Almost periodic solutions for impulsive neural networks with delay. Appl. Math. Model. 31, 1263–1270 (2007)zbMATHCrossRefGoogle Scholar
  12. 12.
    Stamova, I.M.: Stability Analysis of Impulsive Functional Differential Equations. Walter de Gruyter, Berlin (2009)zbMATHCrossRefGoogle Scholar
  13. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Khadra, A., Liu, X., Shen, X.: Impulsive control and synchronization of spatiotemporal chaos. Chaos Solitons Fractals 26, 615–636 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 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)CrossRefGoogle Scholar
  18. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 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)MathSciNetCrossRefGoogle Scholar
  20. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Diethelm, K.: The Analysis of Fractional Differential Equations. An Application-oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin (2010)zbMATHGoogle Scholar
  22. 22.
    Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. Elsevier, New York (2006)zbMATHGoogle Scholar
  23. 23.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  24. 24.
    Babakhani, A., Baleanu, D., Khanbabaie, R.: Hopf bifurcation for a class of fractional differential equations with delay. Nonlinear Dyn. 69, 721–729 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 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)CrossRefGoogle Scholar
  27. 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)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Chen, F., Chen, A., Wang, X.: On the solutions for impulsive fractional functional differential equations. Differ. Equ. Dyn. Syst. 17, 379–391 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Wang, H.: Existence results for fractional functional differential equations with impulses. J. Appl. Math. Comput. 38, 85–101 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  30. 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)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Stamova, I., Stamov, G.: Lipschitz stability criteria for functional differential systems of fractional order. J. Math. Phys. 54, 043502 (2013)MathSciNetCrossRefGoogle Scholar
  32. 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)MathSciNetCrossRefGoogle Scholar
  33. 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)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Li, C., Deng, W., Xu, D.: Chaos synchronization of the Chua system with a fractional order. Physica A 360, 171–185 (2006)MathSciNetCrossRefGoogle Scholar
  35. 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)MathSciNetGoogle Scholar
  36. 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)CrossRefGoogle Scholar
  37. 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)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Huang, X., Zhao, Z., Wang, Z., Li, Y.: Chaos and hyperchaos in fractional-order cellular neural networks. Neurocomputing 94, 13–21 (2012)CrossRefGoogle Scholar
  39. 39.
    Kaslik, E., Sivasundaram, S.: Nonlinear dynamics and chaos in fractional order neural networks. Neural Netw. 32, 245–256 (2012)zbMATHCrossRefGoogle Scholar
  40. 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. 41.
    Yu, J., Hu, C., Jiang, H.: \(\alpha \)-stability and \(\alpha \)-synchronization for fractional-order neural networks. Neural Netw. 35, 82–87 (2012)zbMATHCrossRefGoogle Scholar
  42. 42.
    Zhou, S., Li, H., Zhua, Z.: Chaos control and synchronization in a fractional neuron network system. Chaos Soliton. Fract. 36, 973–984 (2008)zbMATHCrossRefGoogle Scholar
  43. 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)CrossRefGoogle Scholar
  44. 44.
    Wu, R., Hei, X., Chen, L.: Finite-time stability of fractional-order neural networks with delay. Commun. Theor. Phys. 60, 189–193 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  45. 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)zbMATHMathSciNetCrossRefGoogle Scholar
  46. 46.
    Razumikhin, B.S.: Stability of Hereditary Systems. Nauka, Moscow (1988). (in Russian)Google Scholar
  47. 47.
    Yan, J., Shen, J.: Impulsive stabilization of impulsive functional differential equations by Lyapunov-Razumikhin functions. Nonlinear Anal. 37, 245–255 (1999)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of MathematicsThe University of Texas at San AntonioSan AntonioUSA

Personalised recommendations