Generalized Outer Synchronization of Stochastic Neural Networks With Time-Varying Delays

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Abstract

In this paper, generalized outer synchronization between two different stochastic coupled complex dynamical networks with time-varying delays has been investigated. A novel controller is given and the stochastic invariance principle is applied. A stochastic disturbance which is described in term of a Brownian motion are considered in complex dynamical networks. Moreover, some sufficient conditions are derived to ensure generalized outer synchronization of stochastic neural networks. Surprisingly, it is found that complex networks with different structure can be synchronized.

Abstract

In this paper, generalized outer synchronization between two different stochastic coupled complex dynamical networks with time-varying delays has been investigated. A novel controller is given and the stochastic invariance principle is applied. A stochastic disturbance which is described in term of a Brownian motion are considered in complex dynamical networks. Moreover, some sufficient conditions are derived to ensure generalized outer synchronization of stochastic neural networks. Surprisingly, it is found that complex networks with different structure can be synchronized.

References

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  • 2. Balasubramaniam, P.; Lakshmanan, S.; Rakkiyappan, R. - Delay-interval dependent robust stability criteria for stochastic neural networks with linear fractional uncertainties, Neurocomputting, 72 (2009), 3675-3682.

  • 3. Cao, J.D.; Li, L.L. - Cluster synchronization in an array of hybrid coupled neural networks with delay, Neural Networks, 22 (2009), 335-342.

  • 4. Cao, J., Chen, G., Li, P. - Global synchronization in an array of delayed neural networks with hybrid coupling, IEEE Trans. Syst. Man Cybern. B, 38 (2008), 488-498.

  • 5. Cao, J.; Lu, J. - Adaptive synchronization of neural networks with or without time- varying delay, Chaos, 16 (2006), 013133, 6 pp.

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  • 9. Lakshmanan, S.; Balasubramaniam, P. - Linear matrix inequality approach for robust stability analysis for stochastic neural networks with time-varying delay, Chinese Physics B, 20 (2011), 040204.

  • 10. Li, C.P.; Sun, W.G.; Kurths, J. - Synchronization between two coupled complex networks, Physical Review E, 76, Article ID 046204, 2007.

  • 11. Li, C.; Xu, C.; Sun, W.; Xu, J.; Kurths, J. - Outer synchronization of coupled discrete-time networks, Chaos, 19 (2009), 013106, 7 pp.

  • 12. Liang, J.; Wang, Z.; Liu, X. - Robust synchronization of an array of coupled stochastic discrete-time delayed neural networks, IEEE Transactions on Neural Net- works, 9 (2008), 1910-1921.

  • 13. Lu, W.; Chen, T. - Synchronization of coupled connected neural networks with delays, IEEE Trans. Circuits Syst. I Regul. Pap., 51 (2004), 2491-2503.

  • 14. Lu, W.; Chen, T. - Synchronization analysis of linearly coupled networks of discrete time systems, Phys. D, 198 (2004), 148-168.

  • 15. L¨u, J.; Yu, X.; Chen, G. - Chaos synchronization of general complex dynamical networks, Phys. A, 334 (2004), 281-302.

  • 16. Mao, X. - LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl., 236 (1999), 350-369.

  • 17. Rakkiyappan, R.; Balasubramaniam, P.; Lakshmanan, S. - Robust stability results for uncertain stochastic neural networks with discrete interval and distributed time-varying delays, Phys. Lett. A, 372 (2008), 5290-5298.

  • 18. Sun, Y.; Li, W.; Ruan, J. - Generalized outer synchronization between complex dynamical networks with time delay and noise perturbation, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 989-998.

  • 19. Tang, Y.; Wang, Z.; Fang, J. - Pinning control of fractional-order weighted complex networks, Chaos, 19 (2009), 013112, 9 pp.

  • 20. Tang, Y.; Qiu, R.; Fang, J.; Miao, Q.; Xia, M. - Adaptive lag synchronization in unknown stochastic chaotic neural networks with discrete and distributed time-varying delays, Phys. Lett. A, 372 (2008), 4425-4433.

  • 21. Tang, H.; Chen, L.; Lu, J.; Tse, C.K. - Adaptive synchronization between two complex networks with nonidentical topological structures, Physica A, 387 (2008), 5623-5630.

  • 22. Wang, Z.; Wang, Y.; Liu, Y. - Global synchronization for discrete-time stochastic complex networks with randomly occurred nonlinearities and mixed time delays, IEEE Trans. Neural Net., 21 (2010), 11-25.

  • 23. Wang, Y.; Wang, Z.; Liang, J. - A delay fractioning approach to global synchronization of delayed complex networks with stochastic disturbances, Phys. Lett. A, 372 (2008), 6066-6073.

  • 24. Wu, X.; Zheng, W.X.; Zhou, J. - Generalized outer synchronization between complex dynamical networks, Chaos, 19 (2009), 013109, 9 pp.

  • 25. Wu, C.W. - Synchronization in arrays of coupled nonlinear systems with delay and nonreciprocal time-varying coupling, IEEE Transactions on Circuits and Systems II: Express Briefs, 52 (2005), 282-286.

  • 26. Yang, X.; Cao, J. - Stochastic synchronization of coupled neural networks with intermittent control, Phys. Lett. A, 373 (2009), 3259-3272.

  • 27. Yu, W.; Cao, J.; Lü, J. - Global synchronization of linearly hybrid coupled networks with time-varying delay, SIAM J. Appl. Dyn. Syst., 7 (2008), 108-133.

1. Balasubramaniam, P.; Lakshmanan, S. - Delay-interval-dependent robust- stability criteria for neutral stochastic neural networks with polytopic and linear fractional uncertainties, Int. J. Comput. Math., 88 (2011), 2001-2015.

2. Balasubramaniam, P.; Lakshmanan, S.; Rakkiyappan, R. - Delay-interval dependent robust stability criteria for stochastic neural networks with linear fractional uncertainties, Neurocomputting, 72 (2009), 3675-3682.

3. Cao, J.D.; Li, L.L. - Cluster synchronization in an array of hybrid coupled neural networks with delay, Neural Networks, 22 (2009), 335-342.

4. Cao, J., Chen, G., Li, P. - Global synchronization in an array of delayed neural networks with hybrid coupling, IEEE Trans. Syst. Man Cybern. B, 38 (2008), 488-498.

5. Cao, J.; Lu, J. - Adaptive synchronization of neural networks with or without time- varying delay, Chaos, 16 (2006), 013133, 6 pp.

6. Cao, J.D.; Li, P.; Wang, W.W. - Global synchronization in arrays of delayed neural networks with constant and delayed coupling, Physics Letters A, 353 (2006), 318-325.

7. Chen, G.; Zhou, J.; Liu, Z. - Global synchronization of coupled delayed neural networks and applications to chaotic CNN models, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14 (2004), 2229-2240.

8. Gao, H.; Lam, J.; Chen, G. - New criteria for synchronization stability of general complex dynamical networks with coupling delays, Physics Letters A, 360 (2006), 263-273.

9. Lakshmanan, S.; Balasubramaniam, P. - Linear matrix inequality approach for robust stability analysis for stochastic neural networks with time-varying delay, Chinese Physics B, 20 (2011), 040204.

10. Li, C.P.; Sun, W.G.; Kurths, J. - Synchronization between two coupled complex networks, Physical Review E, 76, Article ID 046204, 2007.

11. Li, C.; Xu, C.; Sun, W.; Xu, J.; Kurths, J. - Outer synchronization of coupled discrete-time networks, Chaos, 19 (2009), 013106, 7 pp.

12. Liang, J.; Wang, Z.; Liu, X. - Robust synchronization of an array of coupled stochastic discrete-time delayed neural networks, IEEE Transactions on Neural Net- works, 9 (2008), 1910-1921.

13. Lu, W.; Chen, T. - Synchronization of coupled connected neural networks with delays, IEEE Trans. Circuits Syst. I Regul. Pap., 51 (2004), 2491-2503.

14. Lu, W.; Chen, T. - Synchronization analysis of linearly coupled networks of discrete time systems, Phys. D, 198 (2004), 148-168.

15. L¨u, J.; Yu, X.; Chen, G. - Chaos synchronization of general complex dynamical networks, Phys. A, 334 (2004), 281-302.

16. Mao, X. - LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl., 236 (1999), 350-369.

17. Rakkiyappan, R.; Balasubramaniam, P.; Lakshmanan, S. - Robust stability results for uncertain stochastic neural networks with discrete interval and distributed time-varying delays, Phys. Lett. A, 372 (2008), 5290-5298.

18. Sun, Y.; Li, W.; Ruan, J. - Generalized outer synchronization between complex dynamical networks with time delay and noise perturbation, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 989-998.

19. Tang, Y.; Wang, Z.; Fang, J. - Pinning control of fractional-order weighted complex networks, Chaos, 19 (2009), 013112, 9 pp.

20. Tang, Y.; Qiu, R.; Fang, J.; Miao, Q.; Xia, M. - Adaptive lag synchronization in unknown stochastic chaotic neural networks with discrete and distributed time-varying delays, Phys. Lett. A, 372 (2008), 4425-4433.

21. Tang, H.; Chen, L.; Lu, J.; Tse, C.K. - Adaptive synchronization between two complex networks with nonidentical topological structures, Physica A, 387 (2008), 5623-5630.

22. Wang, Z.; Wang, Y.; Liu, Y. - Global synchronization for discrete-time stochastic complex networks with randomly occurred nonlinearities and mixed time delays, IEEE Trans. Neural Net., 21 (2010), 11-25.

23. Wang, Y.; Wang, Z.; Liang, J. - A delay fractioning approach to global synchronization of delayed complex networks with stochastic disturbances, Phys. Lett. A, 372 (2008), 6066-6073.

24. Wu, X.; Zheng, W.X.; Zhou, J. - Generalized outer synchronization between complex dynamical networks, Chaos, 19 (2009), 013109, 9 pp.

25. Wu, C.W. - Synchronization in arrays of coupled nonlinear systems with delay and nonreciprocal time-varying coupling, IEEE Transactions on Circuits and Systems II: Express Briefs, 52 (2005), 282-286.

26. Yang, X.; Cao, J. - Stochastic synchronization of coupled neural networks with intermittent control, Phys. Lett. A, 373 (2009), 3259-3272.

27. Yu, W.; Cao, J.; Lü, J. - Global synchronization of linearly hybrid coupled networks with time-varying delay, SIAM J. Appl. Dyn. Syst., 7 (2008), 108-133.

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