{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,5]],"date-time":"2026-03-05T13:42:01Z","timestamp":1772718121486,"version":"3.50.1"},"reference-count":24,"publisher":"World Scientific Pub Co Pte Lt","issue":"11","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2012,11]]},"abstract":"<jats:p> The paper presents a detailed analysis on the collective dynamics and delayed state feedback control of a three-dimensional delayed small-world network. The trivial equilibrium of the model is first investigated, showing that the uncontrolled model exhibits complicated unbounded behavior. Then three control strategies, namely a position feedback control, a velocity feedback control, and a hybrid control combined velocity with acceleration feedback, are then introduced to stabilize this unstable system. It is shown in these three control schemes that only the hybrid control can easily stabilize the 3-D network system. And with properly chosen delay and gain in the delayed feedback path, the hybrid controlled model may have stable equilibrium, or periodic solutions resulting from the Hopf bifurcation, or complex stranger attractor from the period-doubling bifurcation. Moreover, the direction of Hopf bifurcation and stability of the bifurcation periodic solutions are analyzed. The results are further extended to any \"d\" dimensional network. It shows that to stabilize a \"d\" dimensional delayed small-world network, at least a \"d \u2013 1\" order completed differential feedback is needed. This work provides a constructive suggestion for the high dimensional delayed systems. <\/jats:p>","DOI":"10.1142\/s0218127412502811","type":"journal-article","created":{"date-parts":[[2012,12,13]],"date-time":"2012-12-13T09:14:17Z","timestamp":1355390057000},"page":"1250281","source":"Crossref","is-referenced-by-count":5,"title":["COLLECTIVE DYNAMICS AND CONTROL OF A 3-D SMALL-WORLD NETWORK WITH TIME DELAYS"],"prefix":"10.1142","volume":"22","author":[{"given":"XU","family":"XU","sequence":"first","affiliation":[{"name":"College of Mathematics, Jilin University, 2699 Qianjin Street, Changchun 130012, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"JIAWEI","family":"LUO","sequence":"additional","affiliation":[{"name":"College of Mathematics, Jilin University, 2699 Qianjin Street, Changchun 130012, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"YUANTONG","family":"GU","sequence":"additional","affiliation":[{"name":"School of Engineering Systems, Queensland University of Technology, QLD 4001, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2012,12,13]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.92.144101"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1016\/j.neucom.2011.03.014"},{"key":"rf4","volume-title":"Theory and Applications of Hopf Bifurcation","author":"Hassard B. 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