{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,19]],"date-time":"2026-07-19T08:22:59Z","timestamp":1784449379458,"version":"3.55.0"},"reference-count":40,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,10,6]],"date-time":"2023-10-06T00:00:00Z","timestamp":1696550400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Natural Science Foundation of China","award":["62071411"],"award-info":[{"award-number":["62071411"]}]},{"name":"National Natural Science Foundation of China","award":["20B567"],"award-info":[{"award-number":["20B567"]}]},{"name":"Research Foundation of Education Department of Hunan Province, China","award":["62071411"],"award-info":[{"award-number":["62071411"]}]},{"name":"Research Foundation of Education Department of Hunan Province, China","award":["20B567"],"award-info":[{"award-number":["20B567"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Recently, the application of memristors in chaotic systems has been extensively studied. Unfortunately, there is limited literature on the introduction of discrete memristors into chaotic maps, especially into non-classical multidimensional maps. For this reason, this paper establishes a new three-dimensional parabolic chaotic map model; in order to improve the complexity and randomness of the map, it is coupled with a square-charge-controlled discrete memristor to design a new four-dimensional memristive hyperchaotic map. Firstly, the stability of the two maps is discussed. And their dynamical properties are compared using Lyapunov exponential spectra and bifurcation diagrams. Then, the phase diagram and iteration sequence of the 4D memristive hyperchaotic map are obtained. Meanwhile, we investigate the hyperchaotic states, the transient chaos, state transfer and attractor coexistence phenomena of the four-dimensional memristive map. In particular, the special state transfer phenomenon of switching from a periodic attractor to a quasi-periodic attractor and the special coexistence phenomenon of a quasi-periodic attractor coexisting with a quasi-periodic attractor around fixed points are found, which have not been observed in other systems. Finally, the phase-track diagrams and iterative sequence diagrams of the four-dimensional memristive map are verified on a digital experimental platform, revealing its potential for practical applications.<\/jats:p>","DOI":"10.3390\/sym15101879","type":"journal-article","created":{"date-parts":[[2023,10,6]],"date-time":"2023-10-06T08:29:12Z","timestamp":1696580952000},"page":"1879","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["A Novel Four-Dimensional Memristive Hyperchaotic Map Based on a Three-Dimensional Parabolic Chaotic Map with a Discrete Memristor"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8457-1307","authenticated-orcid":false,"given":"Mengjiao","family":"Wang","sequence":"first","affiliation":[{"name":"School of Automation and Electronic Information, Xiangtan University, Xiangtan 411105, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Luyao","family":"Tong","sequence":"additional","affiliation":[{"name":"School of Physics and Optoelectronic Engineering, Xiangtan University, Xiangtan 411105, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chunlai","family":"Li","sequence":"additional","affiliation":[{"name":"School of Computer Science & School of Cyberspace Science, Xiangtan University, Xiangtan 411105, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xinan","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Electrical, Electronic and Computer Engineering, University of Western Australia, Crawley, WA 6009, Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Herbert Ho-Ching","family":"Iu","sequence":"additional","affiliation":[{"name":"School of Electrical, Electronic and Computer Engineering, University of Western Australia, Crawley, WA 6009, Australia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhijun","family":"Li","sequence":"additional","affiliation":[{"name":"School of Automation and Electronic Information, Xiangtan University, Xiangtan 411105, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.apacoust.2015.10.015","article-title":"Chaotic direct sequence spread spectrum for secure underwater acoustic communication","volume":"104","author":"Shu","year":"2016","journal-title":"Appl. Acoust."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"43428","DOI":"10.1038\/srep43428","article-title":"Encryption key distribution via chaos synchronization","volume":"7","author":"Keuninckx","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"111749","DOI":"10.1016\/j.chaos.2021.111749","article-title":"Distributed sensing via the ensemble spectra of uncoupled electronic chaotic oscillators","volume":"155","author":"Minati","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1888","DOI":"10.1109\/TCSI.2011.2107190","article-title":"Design of Time-Delay Chaotic Electronic Circuits","volume":"58","author":"Buscarino","year":"2011","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"113429","DOI":"10.1016\/j.chaos.2023.113429","article-title":"Discrete fracmemristor-based chaotic map by Grunwald\u2013Letnikov difference and its circuit implementation","volume":"171","author":"Peng","year":"2023","journal-title":"Chaos Solitons Fractals"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"038701","DOI":"10.1088\/1674-1056\/aca601","article-title":"Hopf bifurcation and phase synchronization in memristor-coupled Hindmarsh\u2013Rose and FitzHugh\u2013Nagumo neurons with two time delays","volume":"32","author":"Guo","year":"2023","journal-title":"Chin. Phys. B"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"058701","DOI":"10.1088\/1674-1056\/acb9f7","article-title":"Synchronization coexistence in a Rulkov neural network based on locally active discrete memristor","volume":"32","author":"Ma","year":"2023","journal-title":"Chin. Phys. B"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.neucom.2022.12.014","article-title":"A discrete memristive neural network and its application for character recognition","volume":"523","author":"He","year":"2023","journal-title":"Neurocomputing"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1016\/S0960-0779(02)00156-X","article-title":"Chemical chaotic schemes derived from NSG system","volume":"15","author":"Xu","year":"2003","journal-title":"Chaos Solitons Fractals"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1109\/TCT.1971.1083337","article-title":"Memristor-the missing circuit element","volume":"18","author":"Chua","year":"1971","journal-title":"IEEE Trans. Circuit Theory"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1016\/j.chaos.2019.01.015","article-title":"A novel memristive 6D hyperchaotic autonomous system with hidden extreme multistability","volume":"120","author":"Mezatio","year":"2019","journal-title":"Chaos Solitons Fractals"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2859","DOI":"10.1007\/s11071-020-05601-x","article-title":"A novel simple chaotic circuit based on memristor\u2013memcapacitor","volume":"100","author":"Ma","year":"2020","journal-title":"Nonlinear Dyn."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1723","DOI":"10.1140\/epjs\/s11734-021-00181-2","article-title":"A simple chaotic circuit with magnetic flux-controlled memristor","volume":"230","author":"Li","year":"2021","journal-title":"Eur. Phys. J. Spec. Top."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2180","DOI":"10.1007\/s11431-011-4400-6","article-title":"Dynamics analysis of chaotic circuit with two memristors","volume":"54","author":"Bao","year":"2011","journal-title":"Sci. China Technol. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"102","DOI":"10.1016\/j.chaos.2016.11.016","article-title":"Hidden extreme multistability in memristive hyperchaotic system","volume":"94","author":"Bao","year":"2017","journal-title":"Chaos Solitons Fractals"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"178","DOI":"10.1016\/j.chaos.2019.03.005","article-title":"Memristor initial boosting behaviors in a two-memristor-based hyperchaotic system","volume":"121","author":"Wu","year":"2019","journal-title":"Chaos Solitons Fractals"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1607","DOI":"10.1007\/s11071-017-3752-2","article-title":"Multistability analysis, circuit implementations and application in image encryption of a novel memristive chaotic circuit","volume":"90","author":"Peng","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1016\/j.cjph.2021.03.009","article-title":"Synchronization, circuit and secure communication implementation of a memristor-based hyperchaotic system using single input controller","volume":"71","author":"Luo","year":"2021","journal-title":"Chin. J. Phys."},{"key":"ref_19","first-page":"2197","article-title":"Two-memristor-based chaotic system with infinite coexisting attractors","volume":"68","author":"Lai","year":"2020","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"769","DOI":"10.1049\/el.2020.1172","article-title":"Hyperchaos in a second-order discrete memristor-based map model","volume":"56","author":"Bao","year":"2020","journal-title":"Electron. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1263","DOI":"10.1007\/s11071-021-06993-0","article-title":"Parameter identification for discrete memristive chaotic map using adaptive differential evolution algorithm","volume":"107","author":"Peng","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1063\/1.5134981","article-title":"Modeling of discrete fracmemristor and its application","volume":"10","author":"He","year":"2020","journal-title":"AIP Adv."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"109873","DOI":"10.1016\/j.chaos.2020.109873","article-title":"A discrete memristor model and its application in H\u00e9non map","volume":"137","author":"Peng","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"153539","DOI":"10.1016\/j.aeue.2020.153539","article-title":"A higher dimensional chaotic map with discrete memristor","volume":"129","author":"Peng","year":"2021","journal-title":"AEU\u2014Int. J. Electron. Commun."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epjp\/s13360-022-02512-1","article-title":"A novel discrete memristive chaotic map","volume":"137","author":"Liang","year":"2022","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Wei, C., Li, G., and Xu, X. (2022). Design of a New Dimension-Changeable Hyperchaotic Model Based on Discrete Memristor. Symmetry, 14.","DOI":"10.3390\/sym14051019"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"111064","DOI":"10.1016\/j.chaos.2021.111064","article-title":"Bifurcation and bursting oscillations in 2D non-autonomous discrete memristor-based hyperchaotic map","volume":"150","author":"Deng","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"4601","DOI":"10.1007\/s11071-021-06544-7","article-title":"Nonparametric bifurcation mechanism in 2-D hyperchaotic discrete memristor-based map","volume":"104","author":"Deng","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"120514","DOI":"10.1088\/1674-1056\/ac20c7","article-title":"Cascade discrete memristive maps for enhancing chaos","volume":"30","author":"Yuan","year":"2021","journal-title":"Chin. Phys. B"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2895","DOI":"10.1007\/s11071-022-07955-w","article-title":"Discretized locally active memristor and application in logarithmic map","volume":"111","author":"Li","year":"2022","journal-title":"Nonlinear Dyn."},{"key":"ref_31","first-page":"2331","article-title":"Design and Implementation of a New Hyperchaotic Memristive Map","volume":"69","author":"Lai","year":"2022","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"key":"ref_32","first-page":"1625","article-title":"A New 3-D Memristive Hyperchaotic Map With Multi-Parameter-Relied Dynamics","volume":"70","author":"Lai","year":"2023","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"4459","DOI":"10.1007\/s11071-022-07380-z","article-title":"Memristive H\u00e9non map with hidden Neimark\u2013Sacker bifurcations","volume":"108","author":"Rong","year":"2022","journal-title":"Nonlinear Dyn."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1016\/j.vlsi.2022.12.003","article-title":"Study of the dynamical behavior of an Ikeda-based map with a discrete memristor","volume":"89","author":"Laskaridis","year":"2023","journal-title":"Integration"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"10504","DOI":"10.7498\/aps.60.010504","article-title":"Dynamics of two-dimensional parabolic discrete map","volume":"60","author":"Meng","year":"2011","journal-title":"Acta Phys. Sin."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"459","DOI":"10.1038\/261459a0","article-title":"Simple mathematical models with very complicated dynamics","volume":"261","author":"May","year":"1976","journal-title":"Nature"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"42","DOI":"10.7498\/aps.71.20211549","article-title":"Simulink modeling and dynamic characteristics of discrete memristor chaotic system","volume":"71","author":"Fu","year":"2022","journal-title":"Acta Phys. Sin."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/j.ijnonlinmec.2019.02.009","article-title":"Dynamics and circuit implementation of a four-wing memristive chaotic system with attractor rotation","volume":"111","author":"Wang","year":"2019","journal-title":"Int. J. Non-Linear Mech."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"3699","DOI":"10.1007\/s11071-021-06763-y","article-title":"Multi-bifurcation cascaded bursting oscillations and mechanism in a novel 3D non-autonomous circuit system with parametric and external excitation","volume":"105","author":"Wang","year":"2021","journal-title":"Nonlinear Dyn."},{"key":"ref_40","unstructured":"Zhang, J.Z. (2010). Nonlinear Dynamic System of Dynamic Stability, Bifurcation Theory and Application, Xi\u2019an Jiaotong University Press (XJUP)."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/10\/1879\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:02:05Z","timestamp":1760130125000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/10\/1879"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,6]]},"references-count":40,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2023,10]]}},"alternative-id":["sym15101879"],"URL":"https:\/\/doi.org\/10.3390\/sym15101879","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,6]]}}}