{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T22:06:23Z","timestamp":1774044383286,"version":"3.50.1"},"reference-count":34,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,2,22]],"date-time":"2025-02-22T00:00:00Z","timestamp":1740182400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This manuscript introduces a simple third-order Hopfield neural network. Its dynamics, implementation with a microcontroller and application to random number generation are explored. The model includes three coupled neurons with no synaptic weights between the first neuron and the third, and between the third and the second. The fundamental features (i.e., symmetry, dissipation and the requirement of existence of an attractor) of the model are studied. The results suggest that the model is asymmetric, dissipative and capable of supporting attractors. The dynamic analysis of the model is conducted through computer explorations, and the findings reveal that it develops complex behaviors like chaos and the coexistence of patterns. The coexistence of patterns is controlled using the linear augmentation method. The coexisting patterns are destroyed, and the multistable system is transformed into a monostable one. In order to confirm the numerical findings, a microcontroller implementation of the considered HNN model is carried out, and the findings of both approaches are concordant. Finally, the elaborated third-order HNN chaotic model is designed for random number generation application. The NIST statistical tests are provided in order to confirm the random features of the generated signals.<\/jats:p>","DOI":"10.3390\/sym17030330","type":"journal-article","created":{"date-parts":[[2025,2,24]],"date-time":"2025-02-24T07:46:57Z","timestamp":1740383217000},"page":"330","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Simple Third-Order Hopfield Neural Network: Dynamic Analysis, Microcontroller Implementation and Application to Random Number Generation"],"prefix":"10.3390","volume":"17","author":[{"given":"Victor Kamdoum","family":"Tamba","sequence":"first","affiliation":[{"name":"Department of Telecommunication and Network Engineering, IUT-Fotso Victor of Bandjoun, University of Dschang, Bandjoun P.O. Box 134, Cameroon"},{"name":"Research Unit of Automation and Applied Computer, Department of Electrical Engineering, IUT-Fotso Victor of Bandjoun, University of Dschang, Bandjoun P.O. Box 134, Cameroon"}]},{"given":"Viet-Thanh","family":"Pham","sequence":"additional","affiliation":[{"name":"Faculty of Electronics Technology, Industrial University of Ho Chi Minh City, Ho Chi Minh City 70000, Vietnam"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8763-7255","authenticated-orcid":false,"given":"Christos","family":"Volos","sequence":"additional","affiliation":[{"name":"Laboratory of Nonlinear Systems, Circuit & Complexity (LaNSCom), Department of Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"522","DOI":"10.1002\/mus.26009","article-title":"Review of the Diagnosis and Treatment of Periodic Paralysis","volume":"57","author":"Statland","year":"2018","journal-title":"Muscle Nerve"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1136\/jnnp-2019-322338","article-title":"Parkinson\u2019s disease: Etiopathogenesis and treatment Journal of Neurology","volume":"91","author":"Jankovic","year":"2020","journal-title":"Neurosurg. 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