{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T20:05:21Z","timestamp":1785441921740,"version":"3.56.0"},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2023,8,24]],"date-time":"2023-08-24T00:00:00Z","timestamp":1692835200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,8,24]],"date-time":"2023-08-24T00:00:00Z","timestamp":1692835200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62003189"],"award-info":[{"award-number":["62003189"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61973189"],"award-info":[{"award-number":["61973189"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2020M672024"],"award-info":[{"award-number":["2020M672024"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2023,10]]},"DOI":"10.1007\/s12190-023-01910-7","type":"journal-article","created":{"date-parts":[[2023,8,24]],"date-time":"2023-08-24T09:02:43Z","timestamp":1692867763000},"page":"3969-3986","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Sampling-based event-triggered control for cluster synchronization in two-layer nonlinear networks"],"prefix":"10.1007","volume":"69","author":[{"given":"Cheng","family":"Zhang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0564-9434","authenticated-orcid":false,"given":"Chuan","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xianfu","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yi","family":"Liang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,8,24]]},"reference":[{"issue":"2","key":"1910_CR1","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1137\/S003614450342480","volume":"45","author":"MEJ Newman","year":"2003","unstructured":"Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45(2), 167\u2013256 (2003)","journal-title":"SIAM Rev."},{"issue":"1","key":"1910_CR2","doi-asserted-by":"publisher","DOI":"10.1063\/1.3077229","volume":"19","author":"S Arianos","year":"2009","unstructured":"Arianos, S., Bompard, E., Carbone, A., Xue, F.: Power grid vulnerability: a complex network approach. Chaos Interdiscipl. J. Nonlinear Sci. 19(1), 013119 (2009)","journal-title":"Chaos Interdiscipl. J. Nonlinear Sci."},{"issue":"1","key":"1910_CR3","first-page":"36","volume":"57","author":"H Huang","year":"2010","unstructured":"Huang, H., Feng, G.: A scaling parameter approach to delay-dependent state estimation of delayed neural networks. IEEE Trans. Circuits Syst. II Express Briefs 57(1), 36\u201340 (2010)","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"issue":"1","key":"1910_CR4","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1016\/j.physd.2006.09.029","volume":"224","author":"A Arenas","year":"2006","unstructured":"Arenas, A., D\u00edaz-Guilera, A., P\u00e9rez-Vicente, C.J.: Synchronization processes in complex networks. Physica D 224(1), 27\u201334 (2006)","journal-title":"Physica D"},{"issue":"1","key":"1910_CR5","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1007\/s12190-020-01346-3","volume":"64","author":"J Zhou","year":"2020","unstructured":"Zhou, J., Bao, H.: Fixed-time synchronization for competitive neural networks with gaussian-wavelet-type activation functions and discrete delays. J. Appl. Math. Comput. 64(1), 103\u2013118 (2020)","journal-title":"J. Appl. Math. Comput."},{"key":"1910_CR6","doi-asserted-by":"publisher","DOI":"10.1109\/TCYB.2022.3228325","author":"S Wu","year":"2022","unstructured":"Wu, S., Li, X.: Finite-time synchronization of complex dynamical networks with input saturation. IEEE Trans. Cybern. (2022). https:\/\/doi.org\/10.1109\/TCYB.2022.3228325","journal-title":"IEEE Trans. Cybern."},{"issue":"5","key":"1910_CR7","doi-asserted-by":"publisher","first-page":"3527","DOI":"10.1007\/s12190-021-01672-0","volume":"68","author":"Y Cheng","year":"2022","unstructured":"Cheng, Y., Zhang, H., Zhang, W., Zhang, H.: Novel algebraic criteria on global Mittag\u2013Leffler synchronization for Foinns with the Caputo derivative and delay. J. Appl. Math. Comput. 68(5), 3527\u20133544 (2022)","journal-title":"J. Appl. Math. Comput."},{"issue":"4","key":"1910_CR8","doi-asserted-by":"publisher","first-page":"154","DOI":"10.1007\/s40314-023-02286-x","volume":"42","author":"J Li","year":"2023","unstructured":"Li, J., Li, H.-L., Yang, J., Yang, J., Zhang, L.: Hybrid control-based synchronization of fractional-order delayed complex-valued fuzzy neural networks. Comput. Appl. Math. 42(4), 154 (2023)","journal-title":"Comput. Appl. Math."},{"issue":"1","key":"1910_CR9","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1016\/0167-2789(94)90274-7","volume":"75","author":"K Kaneko","year":"1994","unstructured":"Kaneko, K.: Relevance of dynamic clustering to biological networks. Physica D 75(1), 55\u201373 (1994)","journal-title":"Physica D"},{"issue":"3","key":"1910_CR10","doi-asserted-by":"publisher","first-page":"262","DOI":"10.1063\/1.166174","volume":"6","author":"NF Rulkov","year":"1996","unstructured":"Rulkov, N.F.: Images of synchronized chaos: experiments with circuits. Chaos: Interdiscipl. J. Nonlinear Sci. 6(3), 262\u2013279 (1996)","journal-title":"Chaos: Interdiscipl. J. Nonlinear Sci."},{"issue":"1","key":"1910_CR11","doi-asserted-by":"publisher","first-page":"87","DOI":"10.3934\/jimo.2011.7.87","volume":"7","author":"X Liu","year":"2011","unstructured":"Liu, X., Chen, T., Lu, W.: Cluster synchronization for linearly coupled complex networks. J. Ind. Manag. Optim. 7(1), 87\u2013101 (2011)","journal-title":"J. Ind. Manag. Optim."},{"issue":"7","key":"1910_CR12","doi-asserted-by":"publisher","first-page":"1009","DOI":"10.1109\/TNN.2011.2139224","volume":"22","author":"X Liu","year":"2011","unstructured":"Liu, X., Chen, T.: Cluster synchronization in directed networks via intermittent pinning control. IEEE Trans. Neural Netw. 22(7), 1009\u20131020 (2011)","journal-title":"IEEE Trans. Neural Netw."},{"issue":"2","key":"1910_CR13","doi-asserted-by":"publisher","first-page":"1107","DOI":"10.1007\/s11071-012-0516-x","volume":"70","author":"J Zhao","year":"2012","unstructured":"Zhao, J., Aziz-Alaoui, M.A., Bertelle, C.: Cluster synchronization analysis of complex dynamical networks by input-to-state stability. Nonlinear Dyn. 70(2), 1107\u20131115 (2012)","journal-title":"Nonlinear Dyn."},{"key":"1910_CR14","doi-asserted-by":"publisher","first-page":"380","DOI":"10.1016\/j.neucom.2022.09.047","volume":"511","author":"H Li","year":"2022","unstructured":"Li, H., Cao, J., Kashkynbayev, A., Cai, S.: Adaptive dynamic event-triggered cluster synchronization in an array of coupled neural networks subject to cyber-attacks. Neurocomputing 511, 380\u2013398 (2022)","journal-title":"Neurocomputing"},{"key":"1910_CR15","doi-asserted-by":"publisher","DOI":"10.1109\/JSYST.2023.3246735","author":"S Zhai","year":"2023","unstructured":"Zhai, S., Wang, X., Zheng, Y.: Cluster synchronization of a nonlinear network with fixed and switching topologies. IEEE Syst. J. (2023). https:\/\/doi.org\/10.1109\/JSYST.2023.3246735","journal-title":"IEEE Syst. J."},{"issue":"11","key":"1910_CR16","doi-asserted-by":"publisher","first-page":"1578","DOI":"10.1049\/cth2.12494","volume":"17","author":"Y Han","year":"2023","unstructured":"Han, Y., Yan, L., Liu, B.: Finite-time cluster synchronization of heterogeneous complex dynamical networks with discontinuous interactions. IET Control Theory Appl. 17(11), 1578\u20131585 (2023)","journal-title":"IET Control Theory Appl."},{"issue":"3","key":"1910_CR17","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.94.031002","volume":"94","author":"C Presigny","year":"2022","unstructured":"Presigny, C., De Vico Fallani, F.: Colloquium: Multiscale modeling of brain network organization. Rev. Mod. Phys. 94(3), 031002 (2022)","journal-title":"Rev. Mod. Phys."},{"issue":"3","key":"1910_CR18","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.81.036118","volume":"81","author":"S Funk","year":"2010","unstructured":"Funk, S., Jansen, V.A.A.: Interacting epidemics on overlay networks. Phys. Rev. E 81(3), 036118 (2010)","journal-title":"Phys. Rev. E"},{"issue":"7","key":"1910_CR19","doi-asserted-by":"publisher","first-page":"2397","DOI":"10.1109\/TCSI.2020.2971608","volume":"67","author":"X Zhang","year":"2020","unstructured":"Zhang, X., Tang, L., L\u00fc, J.: Synchronization analysis on two-layer networks of fractional-order systems: intraiayer and interiayer synchronization. IEEE Trans. Circuits Syst. I Regul. Pap. 67(7), 2397\u20132408 (2020)","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"issue":"1","key":"1910_CR20","doi-asserted-by":"publisher","first-page":"785","DOI":"10.1007\/s11063-020-10281-0","volume":"52","author":"J Zhuang","year":"2020","unstructured":"Zhuang, J., Zhou, Y., Xia, Y.: Intra-layer synchronization in duplex networks with time-varying delays and stochastic perturbations under impulsive control. Neural Process. Lett. 52(1), 785\u2013804 (2020)","journal-title":"Neural Process. Lett."},{"issue":"4","key":"1910_CR21","doi-asserted-by":"publisher","first-page":"1589","DOI":"10.1109\/TCSI.2021.3050988","volume":"68","author":"Y Xu","year":"2021","unstructured":"Xu, Y., Wu, X., Mao, B., L\u00fc, J., Xie, C.: Finite-time intra-layer and inter-layer quasi-synchronization of two-layer multi-weighted networks. IEEE Trans. Circuits Syst. I Regul. Pap. 68(4), 1589\u20131598 (2021)","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"issue":"4","key":"1910_CR22","doi-asserted-by":"publisher","first-page":"2110","DOI":"10.1109\/TCYB.2020.3006032","volume":"52","author":"H Liu","year":"2022","unstructured":"Liu, H., Li, J., Li, Z., Zeng, Z., L\u00fc, J.: Intralayer synchronization of multiplex dynamical networks via pinning impulsive control. IEEE Trans. Cybern. 52(4), 2110\u20132122 (2022)","journal-title":"IEEE Trans. Cybern."},{"issue":"6","key":"1910_CR23","doi-asserted-by":"publisher","first-page":"1458","DOI":"10.3390\/math11061458","volume":"11","author":"C Zhang","year":"2023","unstructured":"Zhang, C., Zhang, C., Meng, F., Liang, Y.: Event-triggered control for intra\/inter-layer synchronization and quasi-synchronization in two-layer coupled networks. Mathematics 11(6), 1458 (2023)","journal-title":"Mathematics"},{"issue":"10","key":"1910_CR24","doi-asserted-by":"publisher","first-page":"3588","DOI":"10.1109\/TAC.2018.2797205","volume":"63","author":"J Shao","year":"2018","unstructured":"Shao, J., Zheng, W.X., Huang, T.-Z., Bishop, A.N.: On leader-follower consensus with switching topologies: an analysis inspired by pigeon hierarchies. IEEE Trans. Autom. Control 63(10), 3588\u20133593 (2018)","journal-title":"IEEE Trans. Autom. Control"},{"issue":"3","key":"1910_CR25","doi-asserted-by":"publisher","first-page":"1239","DOI":"10.1109\/TCNS.2021.3061643","volume":"8","author":"C Shang","year":"2021","unstructured":"Shang, C., Liu, D., Yang, Q., Fang, H.: Responsiveness analysis of classic and observer-based leader-follower systems. IEEE Trans. Control Netw. Syst. 8(3), 1239\u20131248 (2021)","journal-title":"IEEE Trans. Control Netw. Syst."},{"key":"1910_CR26","doi-asserted-by":"publisher","first-page":"264","DOI":"10.1016\/j.neucom.2019.05.077","volume":"359","author":"L Zhou","year":"2019","unstructured":"Zhou, L., Tan, F., Yu, F., Liu, W.: Cluster synchronization of two-layer nonlinearly coupled multiplex networks with multi-links and time-delays. Neurocomputing 359, 264\u2013275 (2019)","journal-title":"Neurocomputing"},{"issue":"4","key":"1910_CR27","first-page":"1338","volume":"68","author":"W Li","year":"2021","unstructured":"Li, W., Zhou, J., Li, J., Xie, T., Lu, J.-A.: Cluster synchronization of two-layer networks via aperiodically intermittent pinning control. IEEE Trans. Circuits Syst. II Express Briefs 68(4), 1338\u20131342 (2021)","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"issue":"2","key":"1910_CR28","doi-asserted-by":"publisher","first-page":"421","DOI":"10.1016\/j.automatica.2009.11.017","volume":"46","author":"E Fridman","year":"2010","unstructured":"Fridman, E.: A refined input delay approach to sampled-data control. Automatica 46(2), 421\u2013427 (2010)","journal-title":"Automatica"},{"issue":"3","key":"1910_CR29","doi-asserted-by":"publisher","first-page":"694","DOI":"10.1002\/acs.3544","volume":"37","author":"W Pan","year":"2023","unstructured":"Pan, W., Zhang, X., Fan, D.: Sampled-data bipartite consensus control of nonlinear multi-agent systems under denial-of-service attacks. Int. J. Adapt. Control Signal Process. 37(3), 694\u2013709 (2023)","journal-title":"Int. J. Adapt. Control Signal Process."},{"issue":"4","key":"1910_CR30","doi-asserted-by":"publisher","first-page":"2913","DOI":"10.1002\/rnc.6542","volume":"33","author":"K Cao","year":"2023","unstructured":"Cao, K., Qian, C., Gu, J.: Global sampled-data stabilization via static output feedback for a class of nonlinear uncertain systems. Int. J. Robust Nonlinear Control 33(4), 2913\u20132929 (2023)","journal-title":"Int. J. Robust Nonlinear Control"},{"issue":"9","key":"1910_CR31","doi-asserted-by":"publisher","first-page":"1680","DOI":"10.1109\/TAC.2007.904277","volume":"52","author":"P Tabuada","year":"2007","unstructured":"Tabuada, P.: Event-triggered real-time scheduling of stabilizing control tasks. IEEE Trans. Autom. Control 52(9), 1680\u20131685 (2007)","journal-title":"IEEE Trans. Autom. Control"},{"key":"1910_CR32","doi-asserted-by":"crossref","unstructured":"Jing, J., Chen, F., Xiang, L.: Event-triggered control for cluster synchronization of linearly coupled complex networks. In: Proceedings of the 33rd Chinese Control Conference, pp. 1299\u20131304 (2014)","DOI":"10.1109\/ChiCC.2014.6896816"},{"key":"1910_CR33","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1016\/j.neunet.2022.03.007","volume":"150","author":"M Wang","year":"2022","unstructured":"Wang, M., Li, X., Duan, P.: Event-triggered delayed impulsive control for nonlinear systems with application to complex neural networks. Neural Netw. 150, 213\u2013221 (2022)","journal-title":"Neural Netw."},{"key":"1910_CR34","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2023.107124","volume":"119","author":"C Zhang","year":"2023","unstructured":"Zhang, C., Zhang, C., Zhang, X., Wang, F., Liang, Y.: Dynamic event-triggered control for intra\/inter-layer synchronization in multi-layer networks. Commun. Nonlinear Sci. Numer. Simul. 119, 107124 (2023)","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"1910_CR35","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.automatica.2019.03.009","volume":"105","author":"C Nowzari","year":"2019","unstructured":"Nowzari, C., Garcia, E., Cort\u00e9s, J.: Event-triggered communication and control of networked systems for multi-agent consensus. Automatica 105, 1\u201327 (2019)","journal-title":"Automatica"},{"issue":"8","key":"1910_CR36","first-page":"3440","volume":"69","author":"K Liu","year":"2022","unstructured":"Liu, K., Ji, Z.: Dynamic event-triggered consensus of general linear multi-agent systems with adaptive strategy. IEEE Trans. Circuits Syst. II Express Briefs 69(8), 3440\u20133444 (2022)","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"issue":"8","key":"1910_CR37","doi-asserted-by":"publisher","first-page":"4328","DOI":"10.1002\/rnc.6610","volume":"33","author":"K Liu","year":"2023","unstructured":"Liu, K., Ji, Z.: Dynamic event-triggered consensus of multi-agent systems under directed topology. Int. J. Robust Nonlinear Control 33(8), 4328\u20134344 (2023)","journal-title":"Int. J. Robust Nonlinear Control"},{"issue":"5","key":"1910_CR38","doi-asserted-by":"publisher","first-page":"1489","DOI":"10.1016\/j.automatica.2014.03.017","volume":"50","author":"G Guo","year":"2014","unstructured":"Guo, G., Ding, L., Han, Q.-L.: A distributed event-triggered transmission strategy for sampled-data consensus of multi-agent systems. Automatica 50(5), 1489\u20131496 (2014)","journal-title":"Automatica"},{"issue":"2","key":"1910_CR39","doi-asserted-by":"publisher","DOI":"10.1063\/1.2184948","volume":"16","author":"Z Ma","year":"2006","unstructured":"Ma, Z., Liu, Z., Zhang, G.: A new method to realize cluster synchronization in connected chaotic networks. Chaos Interdiscipl. J. Nonlinear Sci. 16(2), 023103 (2006)","journal-title":"Chaos Interdiscipl. J. Nonlinear Sci."},{"key":"1910_CR40","doi-asserted-by":"publisher","first-page":"338","DOI":"10.1016\/j.ins.2019.05.049","volume":"512","author":"H L\u00fc","year":"2020","unstructured":"L\u00fc, H., He, W., Han, Q.L., Ge, X., Peng, C.: Finite-time containment control for nonlinear multi-agent systems with external disturbances. Inf. Sci. 512, 338\u2013351 (2020)","journal-title":"Inf. Sci."},{"key":"1910_CR41","doi-asserted-by":"crossref","unstructured":"Zhang, X.M., Han, Q.L.: Stability of linear systems with interval time-varying delays arising from networked control systems. In: IECON 2010 - 36th Annual Conference on IEEE Industrial Electronics Society, pp. 225\u2013230 (2010)","DOI":"10.1109\/IECON.2010.5675203"},{"key":"1910_CR42","doi-asserted-by":"crossref","unstructured":"Boyed, S., Ghaoui, E.l., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM Press, Philadelphia (1994)","DOI":"10.1137\/1.9781611970777"},{"issue":"2","key":"1910_CR43","doi-asserted-by":"publisher","first-page":"475","DOI":"10.1109\/TAC.2012.2206694","volume":"58","author":"D Yue","year":"2013","unstructured":"Yue, D., Tian, E., Han, Q.L.: A delay system method for designing event-triggered controllers of networked control systems. IEEE Trans. Autom. Control 58(2), 475\u2013481 (2013)","journal-title":"IEEE Trans. Autom. Control"}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-023-01910-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-023-01910-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-023-01910-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,26]],"date-time":"2024-10-26T20:05:28Z","timestamp":1729973128000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-023-01910-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,24]]},"references-count":43,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2023,10]]}},"alternative-id":["1910"],"URL":"https:\/\/doi.org\/10.1007\/s12190-023-01910-7","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,24]]},"assertion":[{"value":"21 June 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 August 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 August 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 August 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}