{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T21:25:46Z","timestamp":1778621146485,"version":"3.51.4"},"reference-count":49,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,6,3]],"date-time":"2022-06-03T00:00:00Z","timestamp":1654214400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of China","award":["61901530"],"award-info":[{"award-number":["61901530"]}]},{"name":"Natural Science Foundation of China","award":["62071496"],"award-info":[{"award-number":["62071496"]}]},{"name":"Natural Science Foundation of China","award":["62061008"],"award-info":[{"award-number":["62061008"]}]},{"name":"Natural Science Foundation of China","award":["2020JJ5767"],"award-info":[{"award-number":["2020JJ5767"]}]},{"name":"Natural Science Foundation of Hunan Province","award":["61901530"],"award-info":[{"award-number":["61901530"]}]},{"name":"Natural Science Foundation of Hunan Province","award":["62071496"],"award-info":[{"award-number":["62071496"]}]},{"name":"Natural Science Foundation of Hunan Province","award":["62061008"],"award-info":[{"award-number":["62061008"]}]},{"name":"Natural Science Foundation of Hunan Province","award":["2020JJ5767"],"award-info":[{"award-number":["2020JJ5767"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, we investigate the mathematical models of discrete memristors based on Caputo fractional difference and G\u2013L fractional difference. Specifically, the integer-order discrete memristor is a special model of those two cases. The \u201c\u221e\u201d-type hysteresis loop curves are observed when input is the bipolar periodic signal. Meanwhile, numerical analysis results show that the area of hysteresis decreases with the increase of frequency of input signal and the decrease of derivative order. Moreover, the memory effect, characteristics and physical realization of the discrete memristors are discussed, and a discrete memristor with short memory effects is designed. Furthermore, discrete memristive systems are designed by introducing the fractional-order discrete memristor and integer-order discrete memristor to the Sine map. Chaos is found in the systems, and complexity of the systems is controlled by the parameter of the memristor. Finally, FPGA digital circuit implementation is carried out for the integer-order and fractional-order discrete memristor and discrete memristive systems, which shows the potential application value of the discrete memristor in the engineering application field.<\/jats:p>","DOI":"10.3390\/e24060786","type":"journal-article","created":{"date-parts":[[2022,6,3]],"date-time":"2022-06-03T10:33:01Z","timestamp":1654252381000},"page":"786","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":69,"title":["Discrete Memristor and Discrete Memristive Systems"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5190-4841","authenticated-orcid":false,"given":"Shaobo","family":"He","sequence":"first","affiliation":[{"name":"School of Physics and Electronics, Central South University, Changsha 410083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Donglin","family":"Zhan","sequence":"additional","affiliation":[{"name":"School of Physics and Electronics, Central South University, Changsha 410083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8660-2795","authenticated-orcid":false,"given":"Huihai","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Physics and Electronics, Central South University, Changsha 410083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2503-9262","authenticated-orcid":false,"given":"Kehui","family":"Sun","sequence":"additional","affiliation":[{"name":"School of Physics and Electronics, Central South University, Changsha 410083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6360-9038","authenticated-orcid":false,"given":"Yuexi","family":"Peng","sequence":"additional","affiliation":[{"name":"School of Cyberspace Science, Xiangtan University, Xiangtan 411105, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,3]]},"reference":[{"key":"ref_1","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. 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