Synchronization analysis of hybrid-coupled delayed dynamical networks with impulsive effects: a unified synchronization criterion. (English) Zbl 1395.93224
Summary: This paper addresses the problem of globally exponential synchronization for a class of general hybrid-coupled impulsive delayed dynamical networks with both internal delay and coupling delay. A more general delayed coupling term involving the transmission delay and self-feedback delay is considered. Additionally, two types of impulses occurred in the states of nodes are taken into account: (i) synchronizing impulses defined as they can enhance the synchronization of dynamical networks; and (ii) desynchronizing impulses meaning that they can suppress the synchronization of dynamical networks. By establishing an improved impulsive differential inequality derived based on the average impulsive interval approach, a simpler and less conservative unified globally exponential synchronization criterion is obtained, which is simultaneously effective for synchronizing and desynchronizing impulses. It is shown that the obtained criterion is closely related with impulse strengths, average impulsive interval, and topology structure of the networks. Finally, numerical examples including a typical nearest-neighbor unidirectional coupled network and a scale-free network are given to demonstrate the applicability and efficiency of the theoretical results.
MSC:
| 93B52 | Feedback control |
| 34D06 | Synchronization of solutions to ordinary differential equations |
| 34A37 | Ordinary differential equations with impulses |
| 93C15 | Control/observation systems governed by ordinary differential equations |
| 90B10 | Deterministic network models in operations research |
References:
| [1] | Strogatz, S. H., Exploring complex networks, Nature, 410, 268-276, (2001) · Zbl 1370.90052 |
| [2] | da F. Costa, L.; Oliveira, O. N.; Travieso, G.; Rodrigues, F. A.; Boas, P. R.V.; Antiqueira, L.; Viana, M. P.; Rocha, L. E.C., Analyzing and modeling real-world phenomena with complex networks: a survey of applications, Adv. Phys., 60, 329-412, (2011) |
| [3] | Wang, X.; Chen, G., Complex networks: small-world, scalefree, and beyond, IEEE Circuits Syst. Mag., 3, 6-20, (2003) |
| [4] | Boccaletti, S.; Latora, V.; Moreno, Y.; Chavez, M.; Hwang, D.-U., Complex networks: structure and dynamics, Phys. Rep., 424, 175-308, (2006) · Zbl 1371.82002 |
| [5] | Wu, C. W., Synchronization in Complex Networks of Nonlinear Dynamical Systems, (2007), World Scientific Singapore · Zbl 1135.34002 |
| [6] | Tang, Y.; Gao, H.; Zou, W.; Kurths, J., Distributed synchronization in networks of agent systems with nonlinearities and random switchings, IEEE Trans. Cybern., 43, 358-370, (2013) |
| [7] | Tang, Y.; Gao, H.; Kurths, J., Distributed robust synchronization of dynamical networks with stochastic coupling, IEEE Trans. Circuits Syst. I, 61, 1508-1519, (2014) · Zbl 1468.93188 |
| [8] | Lu, R.; Xu, Y.; Xue, A., \(H_\infty\) filtering for singular systems with communication delays, Signal Process., 90, 1240-1248, (2010) · Zbl 1197.94084 |
| [9] | Lu, R.; Li, H.; Zhu, Y., Quantized \(H_\infty\) filtering for singular time-varying delay systems with unreliable communication channel, Circuits Syst. Signal Process., 31, 521-538, (2012) · Zbl 1253.94008 |
| [10] | Zhou, B.; Li, Z.; Lin, Z., Observer based output feedback control of linear systems with input and output delays, Automatica, 49, 2039-2052, (2013) · Zbl 1364.93309 |
| [11] | Huang, G.; Cao, J., Delay-dependent multistability in recurrent neural networks, Neural Netw., 23, 201-209, (2010) · Zbl 1400.34115 |
| [12] | Zhang, W.; Tang, Y.; Fang, J.; Zhu, W., Exponential cluster synchronization of impulsive delayed genetic oscillators with external disturbances, Chaos, 21, 043137, (2011) · Zbl 1317.34070 |
| [13] | Wong, W. K.; Zhang, W.; Tang, Y.; Wu, X., Stochastic synchronization of complex networks with mixed impulses, IEEE Trans. Circuits Syst. I, 60, 2657-2667, (2013) |
| [14] | Yang, Z.; Xu, D., Stability analysis of delay neural networks with impulsive effects, IEEE Trans. Circuits Syst., 52, 517-521, (2005) |
| [15] | Bainov, D.; Simeonov, P. S., Systems with Impulsive Effect: Stability Theory and Applications, (1989), Ellis Horwood Chichester · Zbl 0676.34035 |
| [16] | Yang, T., Impulsive Control Theory, (2001), Springer Berlin · Zbl 0996.93003 |
| [17] | Lu, J.; Ho, D. W.C.; Cao, J., A unified synchronization criterion for impulsive dynamical networks, Automatica, 46, 1215-1221, (2010) · Zbl 1194.93090 |
| [18] | Liu, B.; Liu, X.; Chen, G.; Wang, H., Robust impulsive synchronization of uncertain dynamical networks, IEEE Trans. Circuits Syst. I, 52, 1431-1441, (2005) · Zbl 1374.82016 |
| [19] | Zhang, G.; Liu, Z.; Ma, Z., Synchronization of complex dynamical networks via impulsive control, Chaos, 17, 043126, (2007) · Zbl 1163.37389 |
| [20] | Cai, S.; Zhou, J.; Xiang, L.; Liu, Z., Robust impulsive synchronization of complex delayed dynamical networks, Phys. Lett. A, 372, 4990-4995, (2008) · Zbl 1221.34075 |
| [21] | Guan, Z.; Liu, Z.; Feng, G.; Wang, Y., Synchronization of complex dynamical networks with time-varying delays via impulsive distributed control, IEEE Trans. Circuits Syst. I, 57, 2182-2195, (2010) · Zbl 1468.93086 |
| [22] | Zhou, J.; Wu, Q.; Xiang, L.; Cai, S.; Liu, Z., Impulsive synchronization seeking in general complex delayed dynamical networks, Nonlinear Anal.Hybrid Sys., 5, 513-524, (2011) · Zbl 1238.93050 |
| [23] | Lu, J.; Kurths, J.; Cao, J.; Mahdavi, N.; Huang, C., Synchronization control for nonlinear stochastic dynamical networks: pinning impulsive strategy, IEEE Trans. Neural Netw., 23, 285-292, (2012) |
| [24] | Yang, X.; Cao, J.; Yang, Z., Synchronization of coupled reaction-diffusion neural networks with time-varying delays via pinning-impulsive controller, SIAM J. Control Optim., 51, 3150-3486, (2013) · Zbl 1281.93052 |
| [25] | Zhou, J.; Xiang, L.; Liu, Z., Synchronization in complex delayed dynamical networks with impulsive effects, Physica A, 384, 684-692, (2007) |
| [26] | B. Liu, D.J. Hill, Robust stability of complex impulsive dynamical systems, in: 46th Proceedings of IEEE Conference on Decision and Control, 2007, pp. 103-108. |
| [27] | Yang, Y.; Cao, J., Exponential synchronization of the complex dynamical networks with a coupling delay and impulsive effects, Nonlinear Anal.: Real World Appl., 11, 1650-1659, (2010) · Zbl 1204.34072 |
| [28] | Lu, J.; Ho, D. W.C.; Cao, J.; Kurths, J., Exponential synchronization of linearly coupled neural networks with impulsive disturbances, IEEE Trans. Neural Netw., 22, 329-335, (2011) |
| [29] | Zhang, Q.; Lu, J.; Lü, J.; Tse, C. K., Adaptive feedback synchronization of a general complex dynamical network with delayed nodes, IEEE Trans. Circuits Syst. II, 55, 183-187, (2008) |
| [30] | Cai, S.; He, Q.; Hao, J.; Liu, Z., Exponential synchronization of complex networks with nonidentical time-delayed dynamical nodes, Phys. Lett. A, 374, 2539-2550, (2010) · Zbl 1236.05185 |
| [31] | Wang, Z.; Zhang, H., Synchronization stability in complex interconnected neural networks with nonsymmetric coupling, Neurocomputing, 108, 84-92, (2013) |
| [32] | 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, 488-498, (2008) |
| [33] | Du, Y.; Xu, R., Robust synchronization of an array of neural networks with hybrid coupling and mixed time delays, ISA Trans., 53, 1015-1023, (2014) |
| [34] | Li, C.; Chen, G., Synchronization in general complex dynamical networks with coupling delays, Physica A, 343, 263-278, (2004) |
| [35] | Zhou, J.; Chen, T., Synchronization in general complex delayed dynamical networks, IEEE Trans. Circuits Syst. I, 53, 733-744, (2006) · Zbl 1374.37056 |
| [36] | Cai, S.; Liu, Z.; Xu, F.; Shen, J., Periodically intermittent controlling complex dynamical networks with time-varying delays to a desired orbit, Phys. Lett. A, 373, 3846-3854, (2009) · Zbl 1234.34035 |
| [37] | Oguchi, T.; Nijmeijer, H.; Yamamoto, T., Synchronization in networks of chaotic system with time-delay coupling, Chaos, 18, 037108, (2008) · Zbl 1309.34104 |
| [38] | Lu, W.; Chen, T.; Chen, G., Synchronization analysis of linearly coupled systems described by differential equations with a coupling delay, Physica D, 221, 118-134, (2006) · Zbl 1111.34056 |
| [39] | He, W.; Cao, J., Exponential synchronization of hybrid coupled networks with delayed coupling, IEEE Trans. Neural Netw., 21, 571-583, (2010) |
| [40] | Münz, U.; Papachristodoulou, A.; Allgöwer, F., Delay robustness in consensus problems, Automatica, 8, 1252-1265, (2010) · Zbl 1204.93013 |
| [41] | Yang, X.; Huang, C.; Zhu, Q., Synchronization of switched neural networks with mixed delays via impulsive control, Chaos Solitons Fractals, 44, 817-826, (2011) · Zbl 1268.93013 |
| [42] | Wang, Y.; Xie, L.; Souza, C., Robust control of a class of uncertain nonlinear systems, Syst. Control Lett., 19, 139-149, (1992) · Zbl 0765.93015 |
| [43] | Halaney, A., Differential EquationsStability, Oscillations, Time Lags, (1966), Academic New York · Zbl 0144.08701 |
| [44] | Pan, L.; Cao, J., Anti-periodic solution for delayed cellular neural networks with impulsive effects, Nonlinear Anal., 12, 3014-3027, (2011) · Zbl 1231.34121 |
| [45] | Hu, J.; Liang, J.; Cao, J., Synchronization of hybrid-coupled heterogeneous networks: pinning control and impulsive control schemes, J. Frankl. Inst., 351, 2600-2622, (2014) · Zbl 1372.93016 |
| [46] | Cai, S.; Zhou, P.; Liu, Z., Synchronization analysis of directed complex networks with time-delayed dynamical nodes and impulsive effects, Nonlinear Dyn., 76, 1677-1691, (2014) · Zbl 1314.34115 |
| [47] | Li, Z., Exponential stability of synchronization in asymmetrically coupled dynamical networks, Chaos, 18, 023124, (2008) · Zbl 1307.34081 |
| [48] | Barabási, A.; Albert, R., Emergence of scaling in random networks, Science, 286, 509-512, (1999) · Zbl 1226.05223 |
| [49] | Pan, L.; Cao, J., Exponential stability of impulsive stochastic functional differential equations, J. Math. Anal. Appl., 382, 672-685, (2011) · Zbl 1222.60043 |
| [50] | Wang, K.; Fu, X.; Li, K., Cluster synchronization in community networks with nonidentical nodes, Chaos, 19, 023106, (2009) · Zbl 1309.34107 |
| [51] | Cao, J.; Li, L., Cluster synchronization in an array of hybrid coupled neural networks with delay, Neural Netw., 22, 335-342, (2009) · Zbl 1338.93284 |
| [52] | Zhang, J.; Ma, Z.; Zhang, G., Cluster synchronization induced by one-node clusters in networks with asymmetric negative couplings, Chaos, 23, 043128, (2013) · Zbl 1331.34118 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
