Link exponential synchronization of discrete-time complex dynamic networks with infinite distributed delay

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

This paper investigates the link exponential synchronization (LES) problem of a class of discrete-time complex dynamic networks (DTCDNs) which consist of infinite distributed delay, node complex dynamic subsystems (NCDS), and link complex dynamic subsystems (LCDS) with mutual coupling relationships. A mathematical induction method is proposed to derive the global exponential stability criterion of error systems. The obtained LES is described using linear scalar inequalities, which are very simple and easy to verify based on standard software tools such as YALMIP. Finally, it was validated through an illustrative example.

Keywords

discrete-time complex dynamic networks (DTCDNs) / link exponential synchronization (LES) / infinite distributed delays

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[1] G. Chen, Z. Guo, Observer-based distributed control and synchronization analysis of inverter-based nonlinear power systems, Nonlinear Dyn., 99 (2020), 2161-2183. https://doi.org/10.1007/s11071-019-05422-7

[2] J. Schiffer, E. Fridman, R. Ortega, J. Raisch, Stability of a class of delayed port-Hamiltonian systems with application to microgrids with distributed rotational and electronic generation, Automatica, 74 (2016), 71-79. https://doi.org/10.1016/j.automatica.2016.07.029

[3] P. Zhu, L. Cheng, C. Gao, Z. Wang, X. Li, Locating multi-sources in social networks with a low infection rate, IEEE Trans. Network Sci. Eng., 9 (2022), 1853-1865. https://doi.org/10.1109/TNSE.2021.3137235

[4] Z. Lv, S. Zhang, W. Xiu, Solving the security problem of intelligent transportation system with deep learning, IEEE Trans. Intell. Transp. Syst., 22 (2021), 4281-4290. https://doi.org/10.1109/TITS.2020.3002185

[5] D. J. Watts, S. H. Strogatz, Collective dynamics of ‘small-world’ networks, Nature, 393 (1998), 440-442. https://doi.org/10.1038/30918

[6] S. H. Strogatz, Exploring complex networks, Nature, 410 (2001), 268-276. https://doi.org/10.1038/35065725

[7] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D. U. Hwang, Complex networks: Structure and dynamics, Phys. Rep., 424 (2006), 175-308. https://doi.org/10.1016/j.physrep.2005.10.009

[8] J. Zhao, Y. Wang, P. Gao, Y. Peng, S. Li, Asymptotical tracking control for the complex network based on the dynamic topology, ISA Trans., 148 (2024), 105-113. https://doi.org/10.1016/j.isatra.2024.02.025

[9] L. Liu, D. Wang, Z. Peng, C. L. P. Chen, T. Li, Bounded neural network control for target tracking of underactuated autonomous surface vehicles in the presence of uncertain target dynamics, IEEE Trans. Neural Networks Learn. Syst., 30 (2019), 1241-1249. https://doi.org/10.1109/TNNLS.2018.2868978

[10] J. Veit, R. Hakim, M. P. Jadi, T. J. Sejnowski, H. Adesnik, Cortical gamma band synchronization through somatostatin interneurons, Nat. Neurosci., 20 (2017), 951-959. https://doi.org/10.1038/nn.4562

[11] M. Bartos, I. Vida, P. Jonas, Synaptic mechanisms of synchronized gamma oscillations in inhibitory interneuron networks, Nat. Rev. Neurosci., 8 (2007), 45-56. https://doi.org/10.1038/nrn2044

[12] P. R. Pagilla, N. B. Siraskar, R. V. Dwivedula, Decentralized control of web processing lines, Trans. Control Syst. Technol., 15 (2006), 106-117. https://doi.org/10.1109/TCST.2006.883345

[13] N. R. Abjadi, J. Soltani, J. Askari, G. R. A. Markadeh, Nonlinear sliding-mode control of a multi-motor web-winding system without tension sensor, Control Theory Appl., 3 (2009), 419-427. https://doi.org/10.1049/iet-cta.2008.0118

[14] H. Fort, M. Scheffer, E. Van Nes, The clumping transition in niche competition: A robust critical phenomenon, J. Stat. Mech: Theory Exp., 2010 (2010), P05005. https://doi.org/10.1088/1742-5468/2010/05/P05005

[15] E. A. Botts, B. F. N. Erasmus, G. J. Alexander, J. Lawlor, Small range size and narrow niche breadth predict range contractions in south african frogs, Global Ecol. Biogeogr., 22 (2013), 567-576. https://doi.org/10.1111/geb.12027

[16] Q. Li, B. Shen, Z. Wang, T. Huang, J. Luo, Synchronization control for a class of discrete time-delay complex dynamical networks: A dynamic event-triggered approach, Trans. Cybern., 49 (2019), 1979-1986. https://doi.org/10.1109/TCYB.2018.2818941

[17] Q. Cheng, J. Cao, Synchronization of complex dynamical networks with discrete time delays on time scales, Neurocomputing, 151 (2015), 729-736. https://doi.org/10.1016/j.neucom.2014.10.033

[18] L. Li, J. Kurths, H. Peng, Y. Yang, Q. Luo, Exponentially asymptotic synchronization of uncertain complex time-delay dynamical networks, Eur. Phys. J. B, 86 (2013), 125. https://doi.org/10.1140/epjb/e2013-30517-6

[19] B. Li, Finite-time synchronization for complex dynamical networks with hybrid coupling and time-varying delay, Nonlinear Dyn., 76 (2014), 1603-1610. https://doi.org/10.1007/s11071-013-1232-x

[20] Y. Liu, Z. Wang, J. Liang, X. Liu, Synchronization and state estimation for discrete-time complex networks with distributed delays, IEEE Trans. Syst. Man Cybern. Part B Cybern., 38 (2008), 1314-1325. https://doi.org/10.1109/TSMCB.2008.925745

[21] Z. Li, J. A. Fang, W. Zhang, X. Wang, Delayed impulsive synchronization of discrete-time complex networks with distributed delays, Nonlinear Dyn., 82 (2015), 2081-2096. https://doi.org/10.1007/s11071-015-2309-5

[22] A. Kazemy, J. Cao, Consecutive synchronization of a delayed complex dynamical network via distributed adaptive control approach, Int. J. Control Autom. Syst., 16 (2018), 2656-2664. https://doi.org/10.1007/s12555-017-0718-6

[23] L. Zhang, Y. Wang, Y. Huang, X. Chen, Delay-dependent synchronization for non-diffusively coupled time-varying complex dynamical networks, Appl. Math. Comput., 259 (2015), 510-522. https://doi.org/10.1016/j.amc.2014.12.034

[24] X. Ruan, J. Feng, C. Xu, J. Wang, Y. Zhao, Dynamic event-triggered pinning synchronization for switched impulsive complex networks with asynchronous switching, IEEE Trans. Circuits Syst. II Express Briefs, 69 (2022), 2211-2215. https://doi.org/10.1109/TCSII.2021.3123285

[25] J. Wang, T. Ru, J. Xia, Y. Wei, Z. Wang, Finite-time synchronization for complex dynamic networks with semi-Markov switching topologies: An H event-triggered control scheme, Appl. Math. Comput., 356 (2019), 235-251. https://doi.org/10.1016/j.amc.2019.03.037

[26] A. Kazemy, K. Shojaei, Synchronization of complex dynamical networks with dynamical behavior links, Asian J. Control, 22 (2020), 474-485. https://doi.org/10.1002/asjc.1910

[27] A. Kazemy, K. Shojaei, Adaptive synchronization of complex dynamical networks in presence of coupling connections with dynamical behavior, J. Comput. Nonlinear Dyn., 14 (2019), 061003. https://doi.org/10.1115/1.4043146

[28] P. Zhao, Y. Wang, Asymptotical stability for complex dynamical networks via link dynamics, Math. Methods Appl. Sci., 43 (2020), 8706-8713. https://doi.org/10.1002/mma.6533

[29] L. Liu, Y. Wang, Z. Gao, Tracking control for the dynamic links of discrete-time complex dynamical network via state observer, Appl. Math. Comput., 369 (2020), 124857. https://doi.org/10.1016/j.amc.2019.124857

[30] P. Gao, Y. Wang, L. Liu, L. Zhang, X. Tang, Asymptotical state synchronization for the controlled directed complex dynamic network via links dynamics, Neurocomputing, 448 (2021), 60-66. https://doi.org/10.1016/j.neucom.2021.03.095

[31] Z. Gao, L. Liu, Y. Wang, P. Gao, Y. Li, Stabilization and synchronization control for complex dynamical networks with dynamic link subsystem, Inf. Sci., 609 (2022), 1588-1600. https://doi.org/10.1016/j.ins.2022.07.153

[32] P. Gao, Y. Wang, J. Zhao, L. Zhang, Y. Peng, Links synchronization control for the complex dynamical network, Neurocomputing, 515 (2023), 59-67. https://doi.org/10.1016/j.neucom.2022.10.024

[33] Y. Wang, W. Wang, L. Zhang, State synchronization of controlled nodes via the dynamics of links for complex dynamical networks, Neurocomputing, 384 (2020), 225-230. https://doi.org/10.1016/j.neucom.2019.12.055

[34] Q. Chen, Y. Peng, Y. Wang, H. Li, Cluster synchronization of delayed complex dynamical networks with the intracluster and intercluster link dynamics, Math. Methods Appl. Sci., 48 (2025), 13542-13551. https://doi.org/10.1002/mma.11121

[35] Q. Chen, Y. Wang, X. Tang, S. Li, Asymptotical outer synchronization control of the complex dynamical networks via unknown interaction and links dynamics, Math. Methods Appl. Sci., 46 (2023), 4379-4390. https://doi.org/10.1002/mma.8765

[36] J. Zhao, Y. Wang, P. Gao, S. Li, Y. Peng, Synchronization of complex dynamical networks with stochastic links dynamics, Entropy, 25 (2023), 1457. https://doi.org/10.3390/e25101457

[37] L. Meng, D. Yang, Q. Zhao, Synchronization of uncertain complex dynamical networks, Int. J. Software Eng. Knowl. Eng., 29 (2019), 923-939. https://doi.org/10.1142/S0218194019400060

[38] L. Zhou, Y. Ma, J. Zou, Robust synchronization of Markovian jump complex dynamical networks under link attacks: Delay-dependent dynamic event-triggered control approach, J. Franklin Inst., 361 (2024), 107251. https://doi.org/10.1016/j.jfranklin.2024.107251

[39] M. Liu, J. Liu, J. Liang, Y. Sun, Y. Shu, Observer-based secure synchronization control of directed complex-valued dynamical networks under link attacks, Nonlinear Dyn., 112 (2024), 12303-12318. https://doi.org/10.1007/s11071-024-09695-5

[40] L. Liu, Y. Wang, H. Chen, Z. Gao, Synchronization control for discrete-time complex dynamical networks with dynamic links subsystem, Mod. Phys. Lett. B, 34 (2020), 2050352. https://doi.org/10.1142/S0217984920503522

[41] L. Liu, Y. Wang, Z. Gao, Tracking control for the connection relationships of discrete-time complex dynamical network associated with the controlled nodes, Int. J. Control Autom. Syst., 17 (2019), 2252-2260. https://doi.org/10.1007/s12555-018-0928-6

[42] L. Liu, Y. Wang, X. Li, Z. Gao, Structural balance for discrete-time complex dynamical network associated with the controlled nodes, Mod. Phys. Lett. B, 34 (2020), 2050098. https://doi.org/10.1142/S0217984920500980

[43] Z. Liang, S. Ding, Y. Jing, X. Xie, Dual-event-driven synchronization control for discrete-time complex dynamical networks, IEEE Trans. Circuits Syst. I-regular papers, 71 (2024), 851-861. https://doi.org/10.1109/TCSI.2023.3341150

[44] Z. Ma, K. Teo, L. Wang, C. Wu, Q. Li, Synchronization of discrete-time complex dynamical networks with delays via impulsive distributed control, Nonlinear Dyn., 114 (2026), 406. https://doi.org/10.1007/s11071-026-12278-1

[45] B. Li, P. Li, Q. Chen, L. Zhong, Y. Wang, Tracking control for discrete-time complex dynamic network via link dynamics, Int. J. Dyn. Control, 13 (2025), 278. https://doi.org/10.1007/s40435-025-01771-x