{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,19]],"date-time":"2025-10-19T15:59:24Z","timestamp":1760889564991,"version":"build-2065373602"},"reference-count":56,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2018,2,13]],"date-time":"2018-02-13T00:00:00Z","timestamp":1518480000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, the synchronization problem of fractional-order complex-valued neural networks with discrete and distributed delays is investigated. Based on the adaptive control and Lyapunov function theory, some sufficient conditions are derived to ensure the states of two fractional-order complex-valued neural networks with discrete and distributed delays achieve complete synchronization rapidly. Finally, numerical simulations are given to illustrate the effectiveness and feasibility of the theoretical results.<\/jats:p>","DOI":"10.3390\/e20020124","type":"journal-article","created":{"date-parts":[[2018,2,13]],"date-time":"2018-02-13T14:23:48Z","timestamp":1518531828000},"page":"124","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Adaptive Synchronization of Fractional-Order Complex-Valued Neural Networks with Discrete and Distributed Delays"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7566-626X","authenticated-orcid":false,"given":"Li","family":"Li","sequence":"first","affiliation":[{"name":"College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7188-5828","authenticated-orcid":false,"given":"Zhen","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China"},{"name":"College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China"}]},{"given":"Junwei","family":"Lu","sequence":"additional","affiliation":[{"name":"School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing 210023, China"}]},{"given":"Yuxia","family":"Li","sequence":"additional","affiliation":[{"name":"College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China"}]}],"member":"1968","published-online":{"date-parts":[[2018,2,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"594","DOI":"10.1016\/j.neucom.2014.08.015","article-title":"Multiple \u03bc-stability analysis of complex-valued neural networks with unbounded time-varying delays","volume":"149","author":"Rakkiyappan","year":"2015","journal-title":"Neurocomputing"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"438","DOI":"10.1002\/cplx.21823","article-title":"Global lagrange stability of complex-valued neural networks of neutral type with time-varying delays","volume":"21","author":"Tu","year":"2016","journal-title":"Complexity"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/j.neucom.2015.06.006","article-title":"Global \u03bc-stability of complex-valued delayed neural networks with leakage delay","volume":"168","author":"Gong","year":"2015","journal-title":"Neurocomputing"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1016\/j.neucom.2015.02.015","article-title":"Stability analysis of complex-valued neural networks with probabilistic time-varying delays","volume":"159","author":"Song","year":"2015","journal-title":"Neurocomputing"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1142\/S0129065795000299","article-title":"Multilayer perceptrons to approximate complex valued functions","volume":"6","author":"Arena","year":"1997","journal-title":"Int. J. Neural Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1016\/S0375-9601(00)00015-3","article-title":"Slow regularization through chaotic oscillation transfer in an unidirectional chain of Hindmarsh-Rose models","volume":"266","author":"Rosa","year":"2000","journal-title":"Phys. Lett. A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1016\/j.neunet.2016.03.007","article-title":"Global exponential stability of complex-valued neural networks with both time-varying delays and impulsive effects","volume":"79","author":"Song","year":"2016","journal-title":"Neural Netw."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1016\/j.neucom.2017.03.015","article-title":"Lagrange stability analysis for complex-valued neural networks with leakage delay and mixed time-varying delays","volume":"244","author":"Song","year":"2017","journal-title":"Neurocomputing"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"4698","DOI":"10.1016\/j.cnsns.2011.03.005","article-title":"Complex valued neural network with M\u00f6bius activation function","volume":"16","year":"2011","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Nitta, T. (2009). Complex-Valued Neural Networks: Utilizing High-Dimensional Parameters, IGI Global.","DOI":"10.4018\/978-1-60566-214-5"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/j.neunet.2016.10.010","article-title":"Dissipativity and stability analysis of fractional-order complex-valued neural networks with time delay","volume":"86","author":"Velmurugan","year":"2017","journal-title":"Neural Netw."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Aizenberg, I. (2011). Complex-Valued Neural Networks with Multi-Valued Neurons, Springer.","DOI":"10.1007\/978-3-642-20353-4"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00521-016-2305-9","article-title":"Multistability of complex-valued neural networks with distributed delays","volume":"28","author":"Gong","year":"2017","journal-title":"Neural Comput. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"10066","DOI":"10.1016\/j.amc.2011.04.085","article-title":"Complex-valued neural network for online complex-valued time-varying matrix inversion","volume":"217","author":"Zhang","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Hirose, A. (2012). Complex-Valued Neural Networks, Springer.","DOI":"10.1007\/978-3-642-27632-3"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1016\/j.amc.2016.07.029","article-title":"Bifurcations in a delayed fractional complex-valued neural network","volume":"292","author":"Huang","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_17","unstructured":"Podlubny, I. (1999). Fractional Differential Equations, Academic Press."},{"key":"ref_18","first-page":"1","article-title":"Monotone iterative method for fractional differential equations","volume":"2016","author":"Bai","year":"2016","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1186\/s13661-016-0573-z","article-title":"Existence results for impulsive nonlinear fractional differential equation with mixed boundary conditions","volume":"2016","author":"Bai","year":"2016","journal-title":"Bound. Value Probl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1531","DOI":"10.1016\/j.camwa.2011.04.057","article-title":"Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay","volume":"62","author":"Wang","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1075","DOI":"10.1016\/j.neucom.2015.07.077","article-title":"Stability analysis of fractional-order Hopfield neural networks with discontinuous activation functions","volume":"171","author":"Zhang","year":"2016","journal-title":"Neurocomputing"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"428","DOI":"10.3182\/20120711-3-BE-2027.00063","article-title":"Entropy and information in a fractional order model of anomalous diffusion","volume":"45","author":"Magin","year":"2012","journal-title":"IFAC Proc. Vol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2317","DOI":"10.1049\/iet-cta.2017.0196","article-title":"Quasi-synchronisation of fractional-order memristor-based neural networks with parameter mismatches","volume":"11","author":"Huang","year":"2017","journal-title":"IET Control Theory Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"6240","DOI":"10.3390\/e16126240","article-title":"Generation and nonlinear dynamical analyses of fractional-order memristor\u2013based Lorenz systems","volume":"16","author":"Xi","year":"2014","journal-title":"Entropy"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1016\/j.physa.2017.04.124","article-title":"Synchronization-based parameter estimation of fractional-order neural networks","volume":"483","author":"Gu","year":"2017","journal-title":"Physica A"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1007\/s11063-017-9604-8","article-title":"Finite-time Stability of Fractional-order Complex-valued Neural Networks with Time Delays","volume":"46","author":"Ding","year":"2017","journal-title":"Neural Process. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2423","DOI":"10.1109\/TNNLS.2016.2574842","article-title":"LMI Conditions for global stability of fractional-order neural networks","volume":"28","author":"Zhang","year":"2016","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/j.neucom.2017.03.042","article-title":"Finite-time stability analysis of fractional-order complex-valued memristor-based neural networks with both leakage and time-varying delays","volume":"245","author":"Wang","year":"2017","journal-title":"Neurocomputing"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"5199","DOI":"10.3390\/e17085199","article-title":"Generalized Combination Complex Synchronization for Fractional-Order Chaotic Complex Systems","volume":"17","author":"Jiang","year":"2015","journal-title":"Entropy"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"296","DOI":"10.1016\/j.amc.2016.11.027","article-title":"Stability analysis of fractional-order complex-valued neural networks with both leakage and discrete delays","volume":"298","author":"Zhang","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1016\/j.neucom.2012.01.011","article-title":"Chaos and hyperchaos in fractional-order cellular neural networks","volume":"94","author":"Huang","year":"2012","journal-title":"Neurocomputing"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.neunet.2013.11.016","article-title":"Global Mittag-Leffler stability and synchronization of memristor-based fractional-order neural networks","volume":"51","author":"Chen","year":"2014","journal-title":"Neural Netw."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/j.neunet.2013.10.002","article-title":"Projective synchronization for fractional neural networks","volume":"49","author":"Yu","year":"2014","journal-title":"Neural Netw."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"6286","DOI":"10.3390\/e16126286","article-title":"Adaptive Synchronization of Fractional Neural Networks with Unknown Parameters and Time Delays","volume":"16","author":"Ma","year":"2014","journal-title":"Entropy"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/j.neunet.2012.02.030","article-title":"Nonlinear dynamics and chaos in fractional-order neural networks","volume":"32","author":"Kaslik","year":"2012","journal-title":"Neural Netw."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.neunet.2016.05.003","article-title":"Synchronization of fractional-order complex-valued neural networks with time delay","volume":"81","author":"Bao","year":"2016","journal-title":"Neural Netw."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1016\/j.amc.2017.12.020","article-title":"H\u221e consensus for nonlinear stochastic multi-agent systems with time delay","volume":"325","author":"Zhou","year":"2018","journal-title":"App. Math. Comput."},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Wang, J., Liang, K., Huang, X., Wang, Z., and Shen, H. (2018). Dissipative fault-tolerant control for nonlinear singular perturbed systems with Markov jumping parameters based on slow state feedback. App. Math. Comput.","DOI":"10.1016\/j.amc.2018.01.049"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"450","DOI":"10.1016\/j.amc.2017.10.039","article-title":"L2 \u2212 L\u221e synchronization for singularly perturbed complex networks with semi-Markov jump topology","volume":"321","author":"Liang","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Wang, Z., Li, L., Li, Y.X., and Cheng, Z.S. (2018). Stability and Hopf bifurcation of a three-neuron network with multiple discrete and distributed delays. Neural Process. Lett.","DOI":"10.1007\/s11063-017-9754-8"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1650271","DOI":"10.1142\/S0217984916502717","article-title":"Dynamical analysis of memristor-based fractional-order neural networks with time delay","volume":"30","author":"Cui","year":"2016","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_42","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_43","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1109\/TNNLS.2014.2311099","article-title":"Existence and uniform stability analysis of fractional-order complex-valued neural networks with time delays","volume":"26","author":"Rakkiyappan","year":"2014","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1750209","DOI":"10.1142\/S0218127417502091","article-title":"Stability and Hopf bifurcation of fractional-order complex-valued single neuron model with time delay","volume":"27","author":"Wang","year":"2017","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1016\/j.chaos.2015.08.003","article-title":"Stability analysis of fractional-order complex-valued neural networks with time delays","volume":"78","author":"Rakkiyappan","year":"2015","journal-title":"Chaos Solitons Fractals"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"2823","DOI":"10.1007\/s11071-014-1628-2","article-title":"Finite-time stability analysis of fractional-order complex-valued memristor-based neural networks with time delays","volume":"78","author":"Rakkiyappan","year":"2014","journal-title":"Nonlinear Dyn."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/00207721.2016.1139761","article-title":"LMI-based approach to stability analysis for fractional-order neural networks with discrete and distributed delays","volume":"49","author":"Zhang","year":"2017","journal-title":"Int. J. Syst. Sci."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.neunet.2017.09.008","article-title":"Synchronization stability of memristor-based complex-valued neural networks with time delays","volume":"96","author":"Liu","year":"2017","journal-title":"Neural Netw."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1007\/s11063-017-9590-x","article-title":"Finite-time synchronization of complex-valued neural networks with mixed delays and uncertain perturbations","volume":"46","author":"Zhou","year":"2017","journal-title":"Neural Process. Lett."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/j.neunet.2017.05.008","article-title":"Master-slave exponential synchronization of delayed complex-valued memristor-based neural networks via impulsive control","volume":"93","author":"Li","year":"2017","journal-title":"Neural Netw."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"3265","DOI":"10.3390\/e15083355","article-title":"Synchronization of a Class of Fractional-Order Chaotic Neural Networks","volume":"15","author":"Chen","year":"2013","journal-title":"Entropy"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1007\/s11063-016-9531-0","article-title":"Stability analysis of fractional order complex-valued memristive neural networks with time delays","volume":"45","author":"Wei","year":"2017","journal-title":"Neural Process. Lett."},{"key":"ref_53","doi-asserted-by":"crossref","unstructured":"Zhang, W.W., Cao, J.D., Chen, D.Y., and Alsaadi, F.E. (2018). Synchronization in fractional-order complex-valued delayed neural networks. Entropy, 20.","DOI":"10.3390\/e20010054"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1343","DOI":"10.1007\/s11071-015-2242-7","article-title":"Adaptive synchronization of fractional-order memristor-based neural networks with time delay","volume":"82","author":"Bao","year":"2015","journal-title":"Nonlinear Dyn."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"2951","DOI":"10.1016\/j.cnsns.2014.01.022","article-title":"Lyapunov functions for fractional order systems","volume":"19","author":"Gallegos","year":"2014","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_56","first-page":"104","article-title":"Mittag-Leffler stability of fractional-order Hopfield neural networks","volume":"16","author":"Zhang","year":"2015","journal-title":"Nonlinear Anal."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/2\/124\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:55:02Z","timestamp":1760194502000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/2\/124"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,2,13]]},"references-count":56,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2018,2]]}},"alternative-id":["e20020124"],"URL":"https:\/\/doi.org\/10.3390\/e20020124","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2018,2,13]]}}}