Abstract
In this paper, we investigate the outer synchronization between two uncertain complex delayed networks with noise coupling. With an adaptive control scheme, sufficient conditions for the stochastic outer synchronization are developed based on the LaSalle invariance principle for stochastic differential equations. Different from existing results, our method is valid for networks with linear or nonlinear inner coupling. Finally, numerical simulations are performed to verify the effectiveness of the proposed synchronization scheme.
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References
Watts, D.J., Strogatz, S.H.: Collective dynamics of small-world networks. Nature 393, 440–442 (1998)
Pecora, L.M., Carroll, T.L.: Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80, 2109–2112 (1998)
Barabási, A.L.: Scale-free networks: a decade and beyond. Science 325, 412–413 (2009)
Adloo, H., Roopaei, M.: Review article on adaptive synchronization of chaotic systems with unknown parameters. Nonlinear Dyn. 65, 141–159 (2011)
Balasubramaniam, P., Vembarasan, V.: Synchronization of recurrent neural networks with mixed time-delays via output coupling with delayed feedback. Nonlinear Dyn. 70, 667–691 (2012)
Zhou, L.Q.: Global asymptotic stability of cellular neural networks with proportional delays. Nonlinear Dyn. 77, 41–47 (2014)
Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)
Wang, X.F., Chen, G.: Synchronization in scale-free dynamical networks: robustness and fragility. IEEE Trans. Circuits Syst. I(49), 54–62 (2002)
Nishikawa, T., Motter, A.E., Lai, Y.C., Hoppensteadt, F.C.: Heterogeneity in oscillator networks: are smaller worlds easier to synchronize? Phys. Rev. Lett. 91, 014101 (2003)
Roopaei, M., Jahromi, M.Z.: Synchronization of two different chaotic systems using novel adaptive fuzzy sliding mode control. Chaos 18, 033133 (2008)
Roopaei, M., Jahromi, M.Z., John, R., Lin, T.C.: Unknown nonlinear chaotic gyros synchronization using adaptive fuzzy sliding mode control with unknown dead-zone input. Commun. Nonlinear Sci. Numer. Simul. 15, 2536–2545 (2010)
Roopaei, M., Jahromi, M.Z., Jafari, S.: Adaptive gain fuzzy sliding mode control for the synchronization of nonlinear chaotic gyros. Chaos 19, 013125 (2009)
Roopaei, M., Zolghadri Jahromi, M.: Synchronization of a class of chaotic systems with fully unknown parameters using adaptive sliding mode approach. Chaos 18, 043112 (2008)
Lü, J.H., Yu, X.H., Chen, G.R.: Chaos synchronization of general complex dynamical networks. Phys. A 334, 281–302 (2004)
Sun, Y.Z., Li, W., Zhao, D.H.: Finite-time stochastic outer synchronization between two complex dynamical networks with different topologies. Chaos 22, 023152 (2012)
Yu, W.W., Chen, G.R., Lü, J.H.: On pinning synchronization of complex dynamical networks. Automatica 45, 429–435 (2009)
Li, C.P., Xu, C.X., Sun, W.G., Kurths, J.: Outer synchronization of coupled discrete-time networks. Chaos 19, 013106 (2009)
Tang, H.W., Chen, L., Lu, J.A., Tse, C.K.: Adaptive synchronization between two complex networks with nonidentical topological structures. Phys. A 387, 5623–5630 (2008)
Wang, G.J., Cao, J.D., Lu, J.Q.: Outer synchronization between two nonidentical networks with circumstance noise. Phys. A 389, 1480–1488 (2010)
Klovdahl, A.S.: Social networks and the spread of infectious diseases: the AIDS example. Soc Sci. Med 21, 1203–1216 (1985)
Begon, M., Townsend, C., Harper, J.: Ecology: Individuals, Populations and Communities. Blackwell Science, London (1996)
Wang, J., Zhang, J., Yuan, Z., Chen, A., Zhou, T.: Neurotransmitter-mediated collective rhythms in grouped Drosophila circadian clocks. J. Biol. Rhythms 23, 472–482 (2008)
Yu, W.W., Cao, J.D., Lü, J.H.: Global synchronization of linearly hybrid coupled networks with time-varying delay. SIAM J. Appl. Dyn. Syst. 7, 108–133 (2008)
Li, P., Yi, Z.: Synchronization analysis of delayed complex networks with time-varying couplings. Phys. A 387, 3729–3737 (2008)
Guo, W.L., Austin, F., Chen, S.H.: Global synchronization of nonlinearly coupled complex networks with non-delayed and delayed coupling. Commun. Nonlinear Sci. Numer. Simul. 15, 1631–1639 (2010)
Sun, Y.Z., Li, W., Ruan, J.: Generalized outer synchronization between complex dynamical networks with time delay and noise perturbation. Commun. Nonlinear Sci. Numer. Simul. 18, 989–998 (2013)
Cai, S.M., Zhou, P.P., Liu, Z.R.: Synchronization analysis of directed complex networks with time-delayed dynamical nodes and impulsive effects. Nonlinear Dyn. 76, 1677–1691 (2014)
Sun, Y.Z., Ruan, J.: Synchronization in coupled time-delayed systems with parameter mismatch and noise perturbation. Chaos 19, 043113 (2009)
Wang, W.P., Li, L.X., Peng, H.P., Yang, Y.X.: Stochastic synchronization of complex network via a novel adaptive nonlinear controller. Nonlinear Dyn. 76, 591–598 (2014)
Neiman, A.B., Russel, F.D.: Synchronization of noise-induced bursts in noncoupled sensory neurons. Phys. Rev. Lett. 88, 138103 (2002)
Hauschildt, B., Jason, N.B., Balanov, A., Scholl, E.: Noise-induced cooperative dynamics and its control in coupled neuron models. Phys. Rev. E 74, 051906 (2006)
Fortuna, L., Frasca, M., Rosa, M.L., Spata, A.: Dynamics of neuron populations in noisy environments. Chaos 15, 014102 (2005)
Korniss, G.: Synchronization in weighted uncorrelated complex networks in a noisy environment: optimization and connections with transport efficiency. Phys. Rev. E 75, 051121 (2007)
Yu, W.W., Cao, J.D.: Synchronization control of stochastic delayed neural networks. Phys. A 373, 252–260 (2007)
Cao, J.D., Wang, Z.D., Sun, Y.H.: Synchronization in an array of linearly stochastically coupled networks with time delay. Phys. A 385, 718–728 (2007)
Hunt, D., Korniss, G., Szymanski, B.K.: Network synchronization in a noisy environment with time delays: fundamental limits and trade-offs. Phys. Rev. Lett. 105, 068701 (2010)
Sun, Y.Z., Shi, H.J., Bakare, E.A., Meng, Q.X.: Noise-induced outer synchronization between two different complex dynamical networks. Nonlinear Dyn. 76, 519–528 (2014)
Nagail, K.H., Kori, H.: Noise-induced synchronization of a large population of globally coupled nonidentical oscillators. Phys. Rev. E 81, 065202 (2010)
Yang, X.S., Cao, J.D., Lu, J.Q.: Synchronization of randomly coupled neural networks with Markovian jum** and time-delay. IEEE Trans. Circuits Syst. I(60), 363–376 (2013)
Zhang, H.G., Gong, D.W., Chen, B., Liu, Z.W.: Synchronization for coupled neural networks with interval delay: a novel augmented LMI method. IEEE Trans. Neural Netw. 24, 58–70 (2013)
Wu, X.J., Lu, H.T.: Outer synchronization of uncertain general complex delayed networks with adaptive coupling. Neurocomputing 82, 157–166 (2012)
Noroozi, N., Roopaei, M., Jahromi, M.Z.: Adaptive fuzzy sliding mode control scheme for uncertain systems. Commun. Nonlinear Sci. Numer. Simul. 14, 3978–3992 (2009)
Noroozi, N., Roopaei, M., Karimaghaee, P., Safavi, A.A.: Simple adaptive variable structure control for unknown chaotic systems. Commun. Nonlinear Sci. Numer. Simul. 15, 707–727 (2010)
Roopaei, M., Sahraei, B.R., Lin, T.C.: Adaptive sliding mode control in a novel class of chaotic systems. Commun. Nonlinear Sci. Numer. Simul. 15, 4158–4170 (2010)
Roopaei, M., Zolghadri, M., Meshksar, S.: Enhanced adaptive fuzzy sliding mode control for uncertain nonlinear systems. Commun. Nonlinear Sci. Numer. Simul. 14, 3670–3681 (2009)
Wang, J., Feng, J., Xu, C., Zhao, Y.: Exponential synchronization of stochastic perturbed complex networks with time-varying delays via periodically intermittent pinning. Commun. Nonlinear Sci. Numer. Simul. 18, 3146–3157 (2013)
Wang, T., Zhou, W., Zhao, S.: Robust synchronization for stochastic delayed complex networks with switching topology and unmodeled dynamics via adaptive control approach. Commun. Nonlinear Sci. Numer. Simul. 18, 2097–2106 (2013)
Mao, X.: LaSalle-type theorems for stochastic differential delay equations. J. Math. Anal. Appl. 236, 350–359 (1999)
Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Heidelberg (1992)
Acknowledgments
We thank anonymous referees for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (Grant No. 61403393) and the Fundamental Research Funds for the Central Universities (Grant No. 2014QNB34).
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Shi, H., Sun, Y., Miao, L. et al. Outer synchronization of uncertain complex delayed networks with noise coupling. Nonlinear Dyn 85, 2437–2448 (2016). https://doi.org/10.1007/s11071-016-2836-8
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DOI: https://doi.org/10.1007/s11071-016-2836-8