{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:13:05Z","timestamp":1760033585290,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2025,5,7]],"date-time":"2025-05-07T00:00:00Z","timestamp":1746576000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Oscillator networks display intricate synchronization patterns. Determining their stability typically requires incorporating the symmetries of the network coupling. Going beyond analyses that appeal only to a network\u2019s automorphism group, we explore synchronization patterns that emerge from the phase-shift invariance of the dynamical equations and symmetries in the nodes. We show that these nonstructural symmetries simplify stability calculations. We analyze a ring-network of phase\u2013amplitude oscillators that exhibits a \u201cdecoupled\u201d state in which physically-coupled nodes appear to act independently due to emergent cancellations in the equations of dynamical evolution. We establish that this state can be linearly stable for a ring of phase\u2013amplitude oscillators, but not for a ring of phase-only oscillators that otherwise require explicit long-range, nonpairwise, or nonphase coupling. In short, amplitude\u2013phase interactions are key to stable synchronization at a distance.<\/jats:p>","DOI":"10.3390\/e27050501","type":"journal-article","created":{"date-parts":[[2025,5,7]],"date-time":"2025-05-07T04:20:26Z","timestamp":1746591626000},"page":"501","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Network and Phase Symmetries Reveal That Amplitude Dynamics Stabilize Decoupled Oscillator Clusters"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2212-1391","authenticated-orcid":false,"given":"Jeffrey","family":"Emenheiser","sequence":"first","affiliation":[{"name":"Complexity Sciences Center and Physics, Mathematics, and Computer Science Departments, University of California, Davis, One Shields Avenue, Davis, CA 95616, USA"}]},{"given":"Anastasiya","family":"Salova","sequence":"additional","affiliation":[{"name":"Complexity Sciences Center and Physics, Mathematics, and Computer Science Departments, University of California, Davis, One Shields Avenue, Davis, CA 95616, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1001-134X","authenticated-orcid":false,"given":"Jordan","family":"Snyder","sequence":"additional","affiliation":[{"name":"Complexity Sciences Center and Physics, Mathematics, and Computer Science Departments, University of California, Davis, One Shields Avenue, Davis, CA 95616, USA"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4466-5410","authenticated-orcid":false,"given":"James P.","family":"Crutchfield","sequence":"additional","affiliation":[{"name":"Complexity Sciences Center and Physics, Mathematics, and Computer Science Departments, University of California, Davis, One Shields Avenue, Davis, CA 95616, USA"}]},{"given":"Raissa M.","family":"D\u2019Souza","sequence":"additional","affiliation":[{"name":"Complexity Sciences Center and Physics, Mathematics, and Computer Science Departments, University of California, Davis, One Shields Avenue, Davis, CA 95616, USA"}]}],"member":"1968","published-online":{"date-parts":[[2025,5,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"037106","DOI":"10.1063\/1.2956986","article-title":"Cluster synchronization in oscillatory networks","volume":"18","author":"Belykh","year":"2008","journal-title":"Chaos Interdiscip. 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