{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T20:02:42Z","timestamp":1775246562690,"version":"3.50.1"},"reference-count":55,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,2,23]],"date-time":"2025-02-23T00:00:00Z","timestamp":1740268800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Guangdong Provincial Key Laboratory","award":["2023B1212060076"],"award-info":[{"award-number":["2023B1212060076"]}]},{"name":"Guangdong Provincial Key Laboratory","award":["6024330003K"],"award-info":[{"award-number":["6024330003K"]}]},{"name":"Guangdong Provincial Key Laboratory","award":["602331006PQ"],"award-info":[{"award-number":["602331006PQ"]}]},{"name":"Guangdong Provincial Key Laboratory","award":["2024KQNC034"],"award-info":[{"award-number":["2024KQNC034"]}]},{"name":"High-Level Talent Initiation Project of Shenzhen Polytechnic University","award":["2023B1212060076"],"award-info":[{"award-number":["2023B1212060076"]}]},{"name":"High-Level Talent Initiation Project of Shenzhen Polytechnic University","award":["6024330003K"],"award-info":[{"award-number":["6024330003K"]}]},{"name":"High-Level Talent Initiation Project of Shenzhen Polytechnic University","award":["602331006PQ"],"award-info":[{"award-number":["602331006PQ"]}]},{"name":"High-Level Talent Initiation Project of Shenzhen Polytechnic University","award":["2024KQNC034"],"award-info":[{"award-number":["2024KQNC034"]}]},{"name":"R&amp;D Center for CNOOC Equipment (Shenzhen) Offshore Engineering Technology, Shenzhen Polytechnic University","award":["2023B1212060076"],"award-info":[{"award-number":["2023B1212060076"]}]},{"name":"R&amp;D Center for CNOOC Equipment (Shenzhen) Offshore Engineering Technology, Shenzhen Polytechnic University","award":["6024330003K"],"award-info":[{"award-number":["6024330003K"]}]},{"name":"R&amp;D Center for CNOOC Equipment (Shenzhen) Offshore Engineering Technology, Shenzhen Polytechnic University","award":["602331006PQ"],"award-info":[{"award-number":["602331006PQ"]}]},{"name":"R&amp;D Center for CNOOC Equipment (Shenzhen) Offshore Engineering Technology, Shenzhen Polytechnic University","award":["2024KQNC034"],"award-info":[{"award-number":["2024KQNC034"]}]},{"name":"Research Projects of Department of Education of Guangdong Province","award":["2023B1212060076"],"award-info":[{"award-number":["2023B1212060076"]}]},{"name":"Research Projects of Department of Education of Guangdong Province","award":["6024330003K"],"award-info":[{"award-number":["6024330003K"]}]},{"name":"Research Projects of Department of Education of Guangdong Province","award":["602331006PQ"],"award-info":[{"award-number":["602331006PQ"]}]},{"name":"Research Projects of Department of Education of Guangdong Province","award":["2024KQNC034"],"award-info":[{"award-number":["2024KQNC034"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Since the research of \u03bc-synchronization helps to explore how complex networks (CNs) work together to produce complex behaviors, the \u03bc-synchronization task for uncertain time-delayed CNs is studied in our work. Especially, bounded external perturbations and variable delays of unknown bounds containing coupling delays, internal delays, and pulse delays are all taken into consideration, making the model more general. Through the \u03bc-stable theory together with the hybrid impulsive control technique, the problems caused by uncertain inner couplings, time-varying delays, and perturbations can be solved, and novel synchronization criteria are gained for the \u03bc-synchronization of the considered CNs. Different from traditional models, it is not necessary for the coupling matrices to meet the zero-row-sum condition, and the control protocol relaxes the constraint of time delays on impulse intervals. Moreover, numerical experiments and image encryption algorithms are carried out to verify our theoretical results\u2019 effectiveness.<\/jats:p>","DOI":"10.3390\/axioms14030161","type":"journal-article","created":{"date-parts":[[2025,2,24]],"date-time":"2025-02-24T05:36:47Z","timestamp":1740375407000},"page":"161","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["New \u03bc-Synchronization Criteria for Nonlinear Drive\u2013Response Complex Networks with Uncertain Inner Couplings and Variable Delays of Unknown Bounds"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9537-101X","authenticated-orcid":false,"given":"Anran","family":"Zhou","sequence":"first","affiliation":[{"name":"College of Computer, Chengdu University, Chengdu 610106, China"},{"name":"Guangdong Provincial Key Laboratory of Intelligent Information Processing and Shenzhen Key Laboratory of Media Security, Shenzhen University, Shenzhen 518060, China"}]},{"given":"Chongming","family":"Yang","sequence":"additional","affiliation":[{"name":"The Training Center, Shenzhen Polytechnic University, Shenzhen 518055, China"}]},{"given":"Chengbo","family":"Yi","sequence":"additional","affiliation":[{"name":"School of Undergraduate Education, Shenzhen Polytechnic University, Shenzhen 518055, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0728-3042","authenticated-orcid":false,"given":"Hongguang","family":"Fan","sequence":"additional","affiliation":[{"name":"College of Computer, Chengdu University, Chengdu 610106, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2224","DOI":"10.1109\/TCYB.2019.2894838","article-title":"A new method for topology identification of complex dynamical networks","volume":"51","author":"Zhu","year":"2021","journal-title":"IEEE Trans. Cybern."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"122475","DOI":"10.1016\/j.eswa.2023.122475","article-title":"Synchronization sampled-data control of uncertain neural networks under an asymmetric Lyapunov-Krasovskii functional method","volume":"239","author":"Wang","year":"2024","journal-title":"Expert Syst. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1539","DOI":"10.1016\/j.automatica.2014.04.012","article-title":"Synchronization in complex networks of phase oscillators: A survey","volume":"50","author":"Dorfler","year":"2014","journal-title":"Automatica"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1016\/j.procs.2011.08.012","article-title":"Phase synchronization approach to construction and analysis of stock correlation network","volume":"6","author":"Sultornsanee","year":"2011","journal-title":"Procedia Comput. Sci."},{"key":"ref_5","first-page":"128872","article-title":"Complete synchronization of delayed discrete-time fractional-order competitive neural networks","volume":"479","author":"Chen","year":"2024","journal-title":"Appl. Math. Comput."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"107057","DOI":"10.1016\/j.cnsns.2022.107057","article-title":"Generalized synchronization of delayed complex-valued dynamical networks via hybrid control","volume":"118","author":"Shen","year":"2023","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3135","DOI":"10.1007\/s00034-015-0194-2","article-title":"Finite-time stability of time-delay switched systems with delayed impulse effects","volume":"35","author":"Gao","year":"2016","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1109\/TCYB.2016.2630703","article-title":"Finite-time and fixed-time cluster synchronization with or without pinning control","volume":"48","author":"Liu","year":"2018","journal-title":"IEEE Trans. Cybern."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Fan, H.G., Chen, X.J., Shi, K.B., Liang, Y.H., Wang, Y., and Wen, H. (2024). Mittag-Leffler synchronization in finite time for uncertain fractional-order multi-delayed memristive neural networks with time-varying perturbations via information feedback. Fractal Fract., 8.","DOI":"10.3390\/fractalfract8070422"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Fan, H.G., Rao, Y., Shi, K.B., and Wen, H. (2024). Time-varying function matrix projection synchronization of Caputo fractional-order uncertain memristive neural networks with multiple delays via mixed open loop feedback control and impulsive control. Fractal Fract., 8.","DOI":"10.3390\/fractalfract8050301"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"115226","DOI":"10.1016\/j.chaos.2024.115226","article-title":"Distributed delayed impulsive control for \u03bc-synchronization of multi-link structure networks with bounded uncertainties and time-varying delays of unmeasured bounds: A novel Halanay impulsive inequality approach","volume":"186","author":"Fan","year":"2024","journal-title":"Chaos Solitons Fractals"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1016\/j.isatra.2023.12.013","article-title":"Global \u03bc-synchronization for coupling delayed complex dynamical networks via event-triggered delayed impulsive control","volume":"145","author":"Zhang","year":"2024","journal-title":"ISA Trans."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"113620","DOI":"10.1016\/j.chaos.2023.113620","article-title":"\u03bc-synchronization of coupled neural networks with hybrid delayed and non-delayed impulsive effects","volume":"173","author":"Fan","year":"2023","journal-title":"Chaos Solitons Fractals"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"498","DOI":"10.1016\/j.neucom.2020.06.022","article-title":"Impulsive \u03bc-stabilization and \u03bc-synchronization for delayed network systems with any time-varying delays","volume":"411","author":"Guan","year":"2020","journal-title":"Neurocomputing"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1016\/j.neunet.2018.07.009","article-title":"Pinning impulsive synchronization for stochastic reaction-diffusion dynamical networks with delay","volume":"106","author":"Chen","year":"2018","journal-title":"Neural Netw."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"129516","DOI":"10.1016\/j.physa.2024.129516","article-title":"Observer-based aperiodically intermittent pinning synchronization of complex-valued dynamical networks with time-varying delay","volume":"635","author":"Liang","year":"2024","journal-title":"Phys. A"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Fan, H.G., Shi, K.B., Guo, Z.Z., and Zhou, A.R. (2024). Finite-time synchronization criteria for Caputo fractional-order uncertain memristive neural networks with fuzzy operators and transmission delay under communication feedback. Fractal Fract., 8.","DOI":"10.3390\/fractalfract8110619"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"127832","DOI":"10.1016\/j.neucom.2024.127832","article-title":"Semiglobal fixed\/preassigned-time synchronization of stochastic neural networks with random delay via adaptive control","volume":"593","author":"Jiang","year":"2024","journal-title":"Neurocomputing"},{"key":"ref_19","first-page":"128834","article-title":"Decentralized control of forward and backward stochastic difference system with nested asymmetric information","volume":"478","author":"Zhang","year":"2024","journal-title":"Appl. Math. Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"107985","DOI":"10.1016\/j.cnsns.2024.107985","article-title":"Fuzzy adaptive event-triggered synchronization control mechanism for T-S fuzzy RDNNs under deception attacks","volume":"134","author":"Wang","year":"2024","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1016\/j.neucom.2022.02.037","article-title":"Finite-time synchronization of the drive-response networks by event-triggered aperiodic intermittent control","volume":"485","author":"Ruan","year":"2022","journal-title":"Neurocomputing"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"124000","DOI":"10.1016\/j.eswa.2024.124000","article-title":"Impulsive synchronization control for dynamic networks subject to double deception attacks","volume":"251","author":"Zhang","year":"2024","journal-title":"Expert Syst. Appl."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/j.neucom.2020.01.035","article-title":"Finite time anti-synchronization of complex-valued neural networks with bounded asynchronous time-varying delays","volume":"387","author":"Liu","year":"2020","journal-title":"Neurocomputing"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1186\/s13662-020-2508-3","article-title":"Finite-time synchronization of uncertain complex dynamic networks with time-varying delays","volume":"32","author":"Luo","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1063","DOI":"10.1109\/JAS.2016.7510043","article-title":"Synchronization of drive-response networks with delays on time scales","volume":"11","author":"Tan","year":"2024","journal-title":"IEEE\/CAA J. Autom. Sin."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1742","DOI":"10.1016\/j.jfranklin.2016.02.003","article-title":"Complex function projective synchronization in drive-response complex-variable dynamical networks with coupling time delays","volume":"353","author":"Han","year":"2016","journal-title":"J. Frankl. Inst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"127288","DOI":"10.1016\/j.neucom.2024.127288","article-title":"Finite-time synchronization of delayed fuzzy inertial neural networks via intermittent control","volume":"574","author":"Wang","year":"2024","journal-title":"Neurocomputing"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"108654","DOI":"10.1016\/j.fss.2023.108654","article-title":"Synchronization and settling-time estimation of fuzzy memristive neural networks with time-varying delays: Fixed-time and preassigned-time control","volume":"470","author":"Wang","year":"2023","journal-title":"Fuzzy Sets Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1760","DOI":"10.1016\/j.isatra.2014.09.022","article-title":"Non-fragile robust synchronization for Markovian jumping chaotic neural networks of neutral-type with randomly occurring uncertainties and mode-dependent time-varying delays","volume":"53","author":"Rakkiyappan","year":"2014","journal-title":"ISA Trans."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1007\/s11424-021-0108-2","article-title":"Stability and Hopf bifurcation analysis of an (n plus m)-neuron double-ring neural network model with multiple time delays","volume":"35","author":"Xing","year":"2022","journal-title":"J. Syst. Sci. Complex."},{"key":"ref_31","first-page":"128110","article-title":"Function matrix projection synchronization for the multi-time delayed fractional order memristor-based neural networks with parameter uncertainty","volume":"454","author":"He","year":"2023","journal-title":"Appl. Math. Comput."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"3657","DOI":"10.1016\/j.jfranklin.2016.06.029","article-title":"Synchronization for fractional-order time-delayed memristor-based neural networks with parameter uncertainty","volume":"353","author":"Gu","year":"2016","journal-title":"J. Frankl. Inst."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"133935","DOI":"10.1016\/j.physd.2023.133935","article-title":"Synchronization of multi-weighted complex networks with mixed variable delays and uncertainties via impulsive pinning control","volume":"456","author":"Fan","year":"2023","journal-title":"Phys. D"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/j.neunet.2019.01.009","article-title":"Disturbance and uncertainty rejection performance for fractional-order complex dynamical networks","volume":"112","author":"Selvaraj","year":"2019","journal-title":"Neural Netw."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"111763","DOI":"10.1016\/j.automatica.2024.111763","article-title":"Uncertainty disturbance estimator control for delayed linear systems with input constraint","volume":"167","author":"Mirshamsi","year":"2024","journal-title":"Automatica"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"6342","DOI":"10.1109\/TSMC.2019.2961753","article-title":"Coordination and control of complex network systems with switching topologies: A survey","volume":"51","author":"Wen","year":"2020","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"ref_37","first-page":"2897","article-title":"Almost sure consensus of multi-agent systems: An intermittent noise","volume":"69","author":"Wu","year":"2022","journal-title":"IEEE Trans. Circuits Syst. II"},{"key":"ref_38","first-page":"287","article-title":"Adaptive modified projective synchronization of gyroscopes with unknown parameters and external disturbances","volume":"12","author":"Tirandaz","year":"2018","journal-title":"ICIC Express Lett."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1443","DOI":"10.1016\/j.neucom.2015.09.018","article-title":"Function projective synchronization in complex dynamical networks with or without external disturbances via error feedback control","volume":"173","author":"Du","year":"2016","journal-title":"Neurocomputing"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"120997","DOI":"10.1016\/j.physa.2019.04.233","article-title":"Modified function projective synchronization between two fractional-order complex dynamical networks with unknown parameters and unknown bounded external disturbances","volume":"526","author":"Du","year":"2019","journal-title":"Phys. A"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"190","DOI":"10.1002\/cplx.21647","article-title":"Mean square exponential synchronization for impulsive coupled neural networks with time-varying delays and stochastic disturbances","volume":"21","author":"Ze","year":"2016","journal-title":"Complexity"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1016\/j.neucom.2021.05.042","article-title":"Synchronization of nonidentical complex dynamical networks with unknown disturbances via observer-based sliding mode control","volume":"454","author":"Zhao","year":"2021","journal-title":"Neurocomputing"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"107227","DOI":"10.1016\/j.cnsns.2023.107227","article-title":"Fixed-time synchronization of switched duplex networks with stochastic disturbances and limited communication","volume":"121","author":"Liang","year":"2023","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"12308","DOI":"10.1016\/j.jfranklin.2020.09.016","article-title":"Fixed-time synchronization of delayed complex dynamical systems with stochastic perturbation via impulsive pinning control","volume":"357","author":"Ren","year":"2020","journal-title":"J. Frankl. Inst."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"015255","DOI":"10.1088\/1402-4896\/ad9c28","article-title":"Mechanical analysis and function matrix projective synchronization of El-Nino chaotic system","volume":"100","author":"Shukla","year":"2025","journal-title":"Phys. Scr."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/j.automatica.2016.06.001","article-title":"Synchronization of linear dynamical networks on time scales: Pinning control via delayed impulses","volume":"72","author":"Liu","year":"2016","journal-title":"Automatica"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"4024","DOI":"10.1109\/TAC.2019.2905271","article-title":"Exponential stability of nonlinear systems with delayed impulses and applications","volume":"64","author":"Li","year":"2019","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_48","first-page":"22","article-title":"Exponential synchronization of complex-valued complex networks with time-varying delays and stochastic perturbations via time-delayed impulsive control","volume":"306","author":"Zhang","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"127484","DOI":"10.1016\/j.physa.2022.127484","article-title":"Global \u03bc-synchronization for nonlinear complex networks with unbounded multiple time delays and uncertainties via impulsive control","volume":"599","author":"Fan","year":"2022","journal-title":"Phys. A"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"123580","DOI":"10.1016\/j.physa.2019.123580","article-title":"Pinning impulsive cluster synchronization of complex dynamical networks","volume":"545","author":"Liu","year":"2020","journal-title":"Phys. A"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.neucom.2020.05.071","article-title":"Leader-following synchronization of complex dynamic networks via event-triggered impulsive control","volume":"412","author":"Peng","year":"2020","journal-title":"Neurocomputing"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/j.neucom.2019.12.097","article-title":"Synchronization of semi-Markov coupled neural networks with impulse effects and leakage delay","volume":"386","author":"Karthick","year":"2020","journal-title":"Neurocomputing"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"227","DOI":"10.3182\/20130703-3-FR-4038.00014","article-title":"Robust synchronization of the structural uncertainty nonlinear network with delay and disturbances","volume":"46","author":"Furtat","year":"2013","journal-title":"IFAC Proc. Vol."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Jia, T.Y., Chen, X.Y., He, L.P., Zhao, F., and Qiu, J.L. (2022). Finite-time synchronization of uncertain fractional-order delayed memristive neural networks via adaptive sliding mode control and its application. Fractal Fract., 6.","DOI":"10.3390\/fractalfract6090502"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.neunet.2015.07.002","article-title":"Impulsive synchronization of Markovian jumping randomly coupled neural networks with partly unknown transition probabilities via multiple integral approach","volume":"70","author":"Chandrasekar","year":"2015","journal-title":"Neural Netw."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/3\/161\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:40:52Z","timestamp":1760028052000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/3\/161"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,23]]},"references-count":55,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["axioms14030161"],"URL":"https:\/\/doi.org\/10.3390\/axioms14030161","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,2,23]]}}}