{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,2]],"date-time":"2026-07-02T11:09:57Z","timestamp":1782990597733,"version":"3.54.5"},"reference-count":43,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,9,30]],"date-time":"2023-09-30T00:00:00Z","timestamp":1696032000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Multidisciplinary Digital Publishing Institute (Switzerland)","award":["0000"],"award-info":[{"award-number":["0000"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We study the Galam majority rule dynamics with contrarian behavior and an oscillating external propaganda in a population of agents that can adopt one of two possible opinions. In an iteration step, a random agent interacts with three other random agents and takes the majority opinion among the agents with probability p(t) (majority behavior) or the opposite opinion with probability 1\u2212p(t) (contrarian behavior). The probability of following the majority rule p(t) varies with the temperature T and is coupled to a time-dependent oscillating field that mimics a mass media propaganda, in a way that agents are more likely to adopt the majority opinion when it is aligned with the sign of the field. We investigate the dynamics of this model on a complete graph and find various regimes as T is varied. A transition temperature Tc separates a bimodal oscillatory regime for T&lt;Tc, where the population\u2019s mean opinion m oscillates around a positive or a negative value from a unimodal oscillatory regime for T&gt;Tc in which m oscillates around zero. These regimes are characterized by the distribution of residence times that exhibit a unique peak for a resonance temperature T*, where the response of the system is maximum. An insight into these results is given by a mean-field approach, which also shows that T* and Tc are closely related.<\/jats:p>","DOI":"10.3390\/e25101402","type":"journal-article","created":{"date-parts":[[2023,10,1]],"date-time":"2023-10-01T16:57:06Z","timestamp":1696179426000},"page":"1402","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Contrarian Majority Rule Model with External Oscillating Propaganda and Individual Inertias"],"prefix":"10.3390","volume":"25","author":[{"given":"Maria Cecilia","family":"Gimenez","sequence":"first","affiliation":[{"name":"Instituto de F\u00edsica Enrique Gaviola (IFEG-Conicet), Facultad de Matem\u00e1tica, Astronom\u00eda, F\u015bica y Computaci\u00f3n (FaMAF, Universidad Nacional de C\u00f3rdoba), C\u00f3rdoba X5000HUA, Argentina"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Luis","family":"Reinaudi","sequence":"additional","affiliation":[{"name":"Instituto de Investigaciones en F\u00edsico-Qu\u00edmica de C\u00f3rdoba (INFIQC, Conicet), Facultad de Ciencias Qu\u00edmicas (Universidad Nacional de C\u00f3rdoba), C\u00f3rdoba X5000HUA, Argentina"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9003-5420","authenticated-orcid":false,"given":"Serge","family":"Galam","sequence":"additional","affiliation":[{"name":"CEVIPOF\u2014Centre for Political Research, Sciences Po and CNRS, 1, Place Saint Thomas d\u2019Aquin, 75007 Paris, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0773-5507","authenticated-orcid":false,"given":"Federico","family":"Vazquez","sequence":"additional","affiliation":[{"name":"Instituto de C\u00e1lculo, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and Conicet, Intendente Guiraldes 2160, Cero + Infinito, Buenos Aires C1428EGA, Argentina"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/0022250X.1982.9989929","article-title":"Sociophysics: A new approach of sociological collective behaviour. 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