{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:32:56Z","timestamp":1760236376279,"version":"build-2065373602"},"reference-count":69,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,11,22]],"date-time":"2021-11-22T00:00:00Z","timestamp":1637539200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11971076"],"award-info":[{"award-number":["11971076"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The classical Hopefield neural networks have obvious symmetry, thus the study related to its dynamic behaviors has been widely concerned. This research article is involved with the neutral-type inertial neural networks incorporating multiple delays. By making an appropriate Lyapunov functional, one novel sufficient stability criterion for the existence and global exponential stability of T-periodic solutions on the proposed system is obtained. In addition, an instructive numerical example is arranged to support the present approach. The obtained results broaden the application range of neutral-types inertial neural networks.<\/jats:p>","DOI":"10.3390\/sym13112231","type":"journal-article","created":{"date-parts":[[2021,12,1]],"date-time":"2021-12-01T03:12:40Z","timestamp":1638328360000},"page":"2231","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Periodicity on Neutral-Type Inertial Neural Networks Incorporating Multiple Delays"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1050-6799","authenticated-orcid":false,"given":"Jian","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114, China"},{"name":"Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha 410114, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ancheng","family":"Chang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Hunan University of Information Technology, Changsha 410151, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gang","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Science, Hunan University of Technology and Business, Changsha 410205, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1016\/0167-2789(86)90152-1","article-title":"Stability and dynamics of simple electronic neural networks with added inertia","volume":"23","author":"Babcock","year":"1986","journal-title":"Physica D"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1016\/0167-2789(87)90021-2","article-title":"Dynamics of simple electronic neural networks","volume":"28","author":"Babcock","year":"1987","journal-title":"Physica D"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Shi, M., Guo, J., Fang, X., and Huang, C. 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