{"title":"开关拓扑下热机械Cucker–Smale模型中单团簇簇的出现","authors":"Jiu‐Gang Dong, Seung‐Yeal Ha, Doheon Kim","doi":"10.1142/s0219530520500025","DOIUrl":null,"url":null,"abstract":"We study the emergent dynamics of the thermomechanical Cucker–Smale (TCS) model with switching network topologies. The TCS model is a generalized CS model with extra internal dynamical variable called “temperature” in which isothermal case exactly coincides with the CS model for flocking. In previous studies, emergent dynamics of the TCS model has been mostly restricted to some static network topologies such as complete graph, connected graph with positive in and out degrees at each node, and digraphs with spanning trees. In this paper, we consider switching network topologies with a spanning tree in a sequence of time-blocks, and present two sufficient frameworks leading to the asymptotic mono-cluster flocking in terms of initial data and system parameters. In the first framework in which the sizes of time-blocks are uniformly bounded by some positive constant, we show that temperature and velocity diameters tend to zero exponentially fast, and spatial diameter is uniformly bounded. In the second framework, we admit a situation in which the sizes of time-blocks may grow mildly by a logarithmic function. In latter framework, our temperature and velocity diameters tend to zero at least algebraically slow.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":"1 1","pages":"1-38"},"PeriodicalIF":2.0000,"publicationDate":"2020-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1142/s0219530520500025","citationCount":"3","resultStr":"{\"title\":\"Emergence of mono-cluster flocking in the thermomechanical Cucker–Smale model under switching topologies\",\"authors\":\"Jiu‐Gang Dong, Seung‐Yeal Ha, Doheon Kim\",\"doi\":\"10.1142/s0219530520500025\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study the emergent dynamics of the thermomechanical Cucker–Smale (TCS) model with switching network topologies. The TCS model is a generalized CS model with extra internal dynamical variable called “temperature” in which isothermal case exactly coincides with the CS model for flocking. In previous studies, emergent dynamics of the TCS model has been mostly restricted to some static network topologies such as complete graph, connected graph with positive in and out degrees at each node, and digraphs with spanning trees. In this paper, we consider switching network topologies with a spanning tree in a sequence of time-blocks, and present two sufficient frameworks leading to the asymptotic mono-cluster flocking in terms of initial data and system parameters. In the first framework in which the sizes of time-blocks are uniformly bounded by some positive constant, we show that temperature and velocity diameters tend to zero exponentially fast, and spatial diameter is uniformly bounded. In the second framework, we admit a situation in which the sizes of time-blocks may grow mildly by a logarithmic function. In latter framework, our temperature and velocity diameters tend to zero at least algebraically slow.\",\"PeriodicalId\":55519,\"journal\":{\"name\":\"Analysis and Applications\",\"volume\":\"1 1\",\"pages\":\"1-38\"},\"PeriodicalIF\":2.0000,\"publicationDate\":\"2020-05-27\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1142/s0219530520500025\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Analysis and Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1142/s0219530520500025\",\"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":"Analysis and Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s0219530520500025","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Emergence of mono-cluster flocking in the thermomechanical Cucker–Smale model under switching topologies
We study the emergent dynamics of the thermomechanical Cucker–Smale (TCS) model with switching network topologies. The TCS model is a generalized CS model with extra internal dynamical variable called “temperature” in which isothermal case exactly coincides with the CS model for flocking. In previous studies, emergent dynamics of the TCS model has been mostly restricted to some static network topologies such as complete graph, connected graph with positive in and out degrees at each node, and digraphs with spanning trees. In this paper, we consider switching network topologies with a spanning tree in a sequence of time-blocks, and present two sufficient frameworks leading to the asymptotic mono-cluster flocking in terms of initial data and system parameters. In the first framework in which the sizes of time-blocks are uniformly bounded by some positive constant, we show that temperature and velocity diameters tend to zero exponentially fast, and spatial diameter is uniformly bounded. In the second framework, we admit a situation in which the sizes of time-blocks may grow mildly by a logarithmic function. In latter framework, our temperature and velocity diameters tend to zero at least algebraically slow.
期刊介绍:
Analysis and Applications publishes high quality mathematical papers that treat those parts of analysis which have direct or potential applications to the physical and biological sciences and engineering. Some of the topics from analysis include approximation theory, asymptotic analysis, calculus of variations, integral equations, integral transforms, ordinary and partial differential equations, delay differential equations, and perturbation methods. The primary aim of the journal is to encourage the development of new techniques and results in applied analysis.