{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T06:40:15Z","timestamp":1777704015267,"version":"3.51.4"},"reference-count":24,"publisher":"SAGE Publications","issue":"3","license":[{"start":{"date-parts":[[2017,8,24]],"date-time":"2017-08-24T00:00:00Z","timestamp":1503532800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Journal of Intelligent &amp; Fuzzy Systems"],"published-print":{"date-parts":[[2017,8,24]]},"abstract":"<jats:p>\n                    Critical dynamics research of recurrent neural networks (RNNs) is very meaningful in both theoretical importance and practical significance. Recently, because of the application requirements, the study on the critical dynamics behaviors of RNNs has drawn special attention. The critical condition is that a discriminant matrix\n                    <jats:italic>M<\/jats:italic>\n                    <jats:sub>1<\/jats:sub>\n                    \u00a0(\n                    <jats:italic>\u0393<\/jats:italic>\n                    ) related with an RNN is nonnegative definite. Due to the essential difficulty in analysis, there were only a few critical results up to now. Further, nearly all of the existing dynamic results are with diagonally nonlinear requirements on the activation mappings, i.e., the activation mapping\n                    <jats:italic>G<\/jats:italic>\n                    should satisfy the strict necessary condition that\n                    <jats:italic>G<\/jats:italic>\n                    \u00a0(\n                    <jats:italic>x<\/jats:italic>\n                    )\u00a0=\u00a0(\n                    <jats:italic>g<\/jats:italic>\n                    <jats:sub>1<\/jats:sub>\n                    \u00a0(\n                    <jats:italic>x<\/jats:italic>\n                    <jats:sub>1<\/jats:sub>\n                    )\u00a0,\n                    <jats:italic>g<\/jats:italic>\n                    <jats:sub>2<\/jats:sub>\n                    \u00a0(\n                    <jats:italic>x<\/jats:italic>\n                    <jats:sub>2<\/jats:sub>\n                    )\u00a0, \u22ef\u00a0,\n                    <jats:italic>g<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>N<\/jats:italic>\n                    <\/jats:sub>\n                    \u00a0(\n                    <jats:italic>x<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>N<\/jats:italic>\n                    <\/jats:sub>\n                    ))\u00a0\n                    <jats:sup>\n                      <jats:italic>T<\/jats:italic>\n                    <\/jats:sup>\n                    . This is because of the essential difficulty on the analysis of the energy function. The requirement is so strict and it limits the applications of RNNs. In this paper, under the critical conditions, some new global asymptotically stable conclusions are presented for RNNs without the diagonally nonlinear requirement on the activation mappings. The results present here not only improve substantially upon the existing relevant critical stability results, but also provide some further cognizance on the essentially dynamical behavior of RNNs, and further, enlarge the application fields of them.\n                  <\/jats:p>","DOI":"10.3233\/jifs-16986","type":"journal-article","created":{"date-parts":[[2017,8,25]],"date-time":"2017-08-25T13:30:06Z","timestamp":1503667806000},"page":"1677-1685","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":0,"title":["Critical dynamical analysis for \u03b1-UAM RNNs without diagonal nonlinear requirements"],"prefix":"10.1177","volume":"33","author":[{"given":"Chen","family":"Qiao","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Xi\u2019an Jiaotong University, Xi\u2019an, P.R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kefeng","family":"Sun","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Xi\u2019an Jiaotong University, Xi\u2019an, P.R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wenfeng","family":"Jing","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Xi\u2019an Jiaotong University, Xi\u2019an, P.R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yan","family":"Shi","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Xi\u2019an Jiaotong University, Xi\u2019an, P.R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2017,8,24]]},"reference":[{"key":"e_1_3_2_2_2","doi-asserted-by":"publisher","DOI":"10.1109\/TNNLS.2014.2317880"},{"issue":"99","key":"e_1_3_2_3_2","first-page":"1","article-title":"Stability of recurrent neural networks with time-varying delay via flexible terminal method","author":"Wang Z.S.","year":"2016","unstructured":"WangZ.S., DingS.B., ShanQ.H. and ZhangH.G., Stability of recurrent neural networks with time-varying delay via flexible terminal method, IEEE Transactions on Neural Networks & Learning Systems PP(99) (2016), 1\u20138.","journal-title":"IEEE Transactions on Neural Networks & Learning Systems"},{"key":"e_1_3_2_4_2","doi-asserted-by":"publisher","DOI":"10.1109\/72.963793"},{"key":"e_1_3_2_5_2","doi-asserted-by":"publisher","DOI":"10.1109\/72.914529"},{"key":"e_1_3_2_6_2","doi-asserted-by":"publisher","DOI":"10.1109\/72.914530"},{"key":"e_1_3_2_7_2","doi-asserted-by":"publisher","DOI":"10.1109\/TCSI.2002.803358"},{"key":"e_1_3_2_8_2","doi-asserted-by":"publisher","DOI":"10.1109\/TNNLS.2016.2574748"},{"key":"e_1_3_2_9_2","doi-asserted-by":"publisher","DOI":"10.1109\/TCYB.2013.2262935"},{"key":"e_1_3_2_10_2","doi-asserted-by":"publisher","DOI":"10.1109\/TCYB.2015.2388582"},{"key":"e_1_3_2_11_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0893-6080(03)00192-8"},{"key":"e_1_3_2_12_2","doi-asserted-by":"publisher","DOI":"10.1109\/TSMCB.2002.804368"},{"key":"e_1_3_2_13_2","doi-asserted-by":"publisher","DOI":"10.1109\/TCSI.2005.844366"},{"key":"e_1_3_2_14_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2013\/760293","article-title":"Without diagonal nonlinear requirements: The more general-critical dynamical analysis for UPPAM recurrent neural networks","volume":"2013","author":"Chen X.","year":"2013","unstructured":"ChenX., MaoH.Z. and QiaoC., Without diagonal nonlinear requirements: The more general-critical dynamical analysis for UPPAM recurrent neural networks, Mathematical Problems in Engineering 2013 (2013), 1\u201310.","journal-title":"Mathematical Problems in Engineering"},{"key":"e_1_3_2_15_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.neucom.2011.09.008"},{"key":"e_1_3_2_16_2","doi-asserted-by":"publisher","DOI":"10.1109\/TCSI.2002.803358"},{"key":"e_1_3_2_17_2","doi-asserted-by":"publisher","DOI":"10.1109\/TNN.2006.880579"},{"key":"e_1_3_2_18_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-72395-0_18"},{"key":"e_1_3_2_19_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.neucom.2008.06.006"},{"key":"e_1_3_2_20_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.neucom.2010.04.008"},{"key":"e_1_3_2_21_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.neucom.2012.10.026"},{"key":"e_1_3_2_22_2","doi-asserted-by":"publisher","DOI":"10.1007\/s11424-015-2053-4"},{"key":"e_1_3_2_23_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.neucom.2014.12.007"},{"key":"e_1_3_2_24_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.neucom.2015.09.103"},{"key":"e_1_3_2_25_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10114-011-0598-2"}],"container-title":["Journal of Intelligent &amp; Fuzzy Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/JIFS-16986","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/full-xml\/10.3233\/JIFS-16986","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/JIFS-16986","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T09:40:13Z","timestamp":1777455613000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/JIFS-16986"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,8,24]]},"references-count":24,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2017,8,24]]}},"alternative-id":["10.3233\/JIFS-16986"],"URL":"https:\/\/doi.org\/10.3233\/jifs-16986","relation":{},"ISSN":["1064-1246","1875-8967"],"issn-type":[{"value":"1064-1246","type":"print"},{"value":"1875-8967","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,8,24]]}}}