{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T02:28:42Z","timestamp":1747189722876,"version":"3.40.5"},"reference-count":14,"publisher":"World Scientific Pub Co Pte Ltd","issue":"16","funder":[{"DOI":"10.13039\/100000006","name":"Office of Naval Research","doi-asserted-by":"crossref","award":["N00014-16-1-2134"],"award-info":[{"award-number":["N00014-16-1-2134"]}],"id":[{"id":"10.13039\/100000006","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2020,12,30]]},"abstract":"<jats:p>Computational and experimental works reveal that the coupling of similar crystal oscillators leads to a variety of collective patterns, mainly various forms of discrete rotating waves and synchronization patterns, which have the potential for developing precision timing devices through phase drift reduction. Among all observed patterns, the standard traveling wave, in which consecutive crystals oscillate out of phase by [Formula: see text], where [Formula: see text] is the network size, leads to optimal phase drift error that scales down as [Formula: see text] as opposed to [Formula: see text] for an uncoupled ensemble. In this manuscript, we provide an analytical proof of the scaling laws, for uncoupled and coupled symmetric networks, and show that [Formula: see text] is the fundamental limit of phase-error reduction that one can obtain with a symmetric network of nonlinear oscillators of any type, not just crystals.<\/jats:p>","DOI":"10.1142\/s0218127420502533","type":"journal-article","created":{"date-parts":[[2020,12,30]],"date-time":"2020-12-30T09:16:23Z","timestamp":1609319783000},"page":"2050253","source":"Crossref","is-referenced-by-count":2,"title":["On the Scaling Law of Phase Drift in Coupled Nonlinear Oscillators for Precision Timing"],"prefix":"10.1142","volume":"30","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0580-3657","authenticated-orcid":false,"given":"Antonio","family":"Palacios","sequence":"first","affiliation":[{"name":"Nonlinear Dynamical Systems Group, Department of Mathematics, San Diego State University, San Diego, CA 92182, USA"}]},{"given":"Pietro-Luciano","family":"Buono","sequence":"additional","affiliation":[{"name":"Universit\u00e9 du Qu\u00e9bec \u00e0 Rimouski, 300, All\u00e9e des Ursulines, Rimouski (Qu\u00e9bec) G5L 3A1, Canada"}]},{"given":"Visarath","family":"In","sequence":"additional","affiliation":[{"name":"Naval Information Warfare Center Pacific, Code 71740, 53560 Hull Street, San Diego, CA 92152-5001, USA"}]},{"given":"Patrick","family":"Longhini","sequence":"additional","affiliation":[{"name":"Naval Information Warfare Center Pacific, Code 71740, 53560 Hull Street, San Diego, CA 92152-5001, USA"}]}],"member":"219","published-online":{"date-parts":[[2020,12,28]]},"reference":[{"key":"S0218127420502533BIB001","series-title":"Mathematics in Science and Engineering","volume-title":"Stochastic Systems","volume":"169","author":"Adomian G.","year":"1983"},{"key":"S0218127420502533BIB002","doi-asserted-by":"crossref","first-page":"1310","DOI":"10.1137\/16M1066154","volume":"17","author":"Buono P.-L.","year":"2018","journal-title":"SIAM J. 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