{"title":"Kuramoto平均场博弈中的同步","authors":"Rene Carmona, Quentin Cormier, H. Mete Soner","doi":"10.1080/03605302.2023.2264611","DOIUrl":null,"url":null,"abstract":"AbstractThe classical Kuramoto model is studied in the setting of an infinite horizon mean field game. The system is shown to exhibit both synchronization and phase transition. Incoherence below a critical value of the interaction parameter is demonstrated by the stability of the uniform distribution. Above this value, the game bifurcates and develops self-organizing time homogeneous Nash equilibria. As interactions get stronger, these stationary solutions become fully synchronized. Results are proved by an amalgam of techniques from nonlinear partial differential equations, viscosity solutions, stochastic optimal control and stochastic processes.KEYWORDS: Mean field gamesKuramoto modelsynchronizationviscosity solutions2020 MATHEMATICS SUBJECT CLASSIFICATION: 35Q8935D4039N8091A1692B25 Additional informationFundingResearch of Carmona was partially supported by AFOSR FA9550-19-1-0291 and ARPA-E DE-AR0001289. Research of Soner was partially supported by the National Science Foundation grant DMS 2106462.","PeriodicalId":50657,"journal":{"name":"Communications in Partial Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.1000,"publicationDate":"2023-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"6","resultStr":"{\"title\":\"Synchronization in a Kuramoto mean field game\",\"authors\":\"Rene Carmona, Quentin Cormier, H. Mete Soner\",\"doi\":\"10.1080/03605302.2023.2264611\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"AbstractThe classical Kuramoto model is studied in the setting of an infinite horizon mean field game. The system is shown to exhibit both synchronization and phase transition. Incoherence below a critical value of the interaction parameter is demonstrated by the stability of the uniform distribution. Above this value, the game bifurcates and develops self-organizing time homogeneous Nash equilibria. As interactions get stronger, these stationary solutions become fully synchronized. Results are proved by an amalgam of techniques from nonlinear partial differential equations, viscosity solutions, stochastic optimal control and stochastic processes.KEYWORDS: Mean field gamesKuramoto modelsynchronizationviscosity solutions2020 MATHEMATICS SUBJECT CLASSIFICATION: 35Q8935D4039N8091A1692B25 Additional informationFundingResearch of Carmona was partially supported by AFOSR FA9550-19-1-0291 and ARPA-E DE-AR0001289. Research of Soner was partially supported by the National Science Foundation grant DMS 2106462.\",\"PeriodicalId\":50657,\"journal\":{\"name\":\"Communications in Partial Differential Equations\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2023-09-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"6\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications in Partial Differential Equations\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1080/03605302.2023.2264611\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Partial Differential Equations","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1080/03605302.2023.2264611","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
AbstractThe classical Kuramoto model is studied in the setting of an infinite horizon mean field game. The system is shown to exhibit both synchronization and phase transition. Incoherence below a critical value of the interaction parameter is demonstrated by the stability of the uniform distribution. Above this value, the game bifurcates and develops self-organizing time homogeneous Nash equilibria. As interactions get stronger, these stationary solutions become fully synchronized. Results are proved by an amalgam of techniques from nonlinear partial differential equations, viscosity solutions, stochastic optimal control and stochastic processes.KEYWORDS: Mean field gamesKuramoto modelsynchronizationviscosity solutions2020 MATHEMATICS SUBJECT CLASSIFICATION: 35Q8935D4039N8091A1692B25 Additional informationFundingResearch of Carmona was partially supported by AFOSR FA9550-19-1-0291 and ARPA-E DE-AR0001289. Research of Soner was partially supported by the National Science Foundation grant DMS 2106462.
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This journal aims to publish high quality papers concerning any theoretical aspect of partial differential equations, as well as its applications to other areas of mathematics. Suitability of any paper is at the discretion of the editors. We seek to present the most significant advances in this central field to a wide readership which includes researchers and graduate students in mathematics and the more mathematical aspects of physics and engineering.
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