{"title":"固定低维状态空间中的任意大型异斜网络","authors":"S. Castro, Alexander Lohse","doi":"10.1063/5.0156192","DOIUrl":null,"url":null,"abstract":"We consider heteroclinic networks between n∈N nodes where the only connections are those linking each node to its two subsequent neighboring ones. Using a construction method where all nodes are placed in a single one-dimensional space and the connections lie in coordinate planes, we show that it is possible to robustly realize these networks in R6 for any number of nodes n using a polynomial vector field. This bound on the space dimension (while the number of nodes in the network goes to ∞) is a novel phenomenon and a step toward more efficient realization methods for given connection structures in terms of the required number of space dimensions. We briefly discuss some stability properties of the generated heteroclinic objects.","PeriodicalId":340975,"journal":{"name":"Chaos: An Interdisciplinary Journal of Nonlinear Science","volume":"16 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2023-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Arbitrarily large heteroclinic networks in fixed low-dimensional state space\",\"authors\":\"S. Castro, Alexander Lohse\",\"doi\":\"10.1063/5.0156192\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider heteroclinic networks between n∈N nodes where the only connections are those linking each node to its two subsequent neighboring ones. Using a construction method where all nodes are placed in a single one-dimensional space and the connections lie in coordinate planes, we show that it is possible to robustly realize these networks in R6 for any number of nodes n using a polynomial vector field. This bound on the space dimension (while the number of nodes in the network goes to ∞) is a novel phenomenon and a step toward more efficient realization methods for given connection structures in terms of the required number of space dimensions. We briefly discuss some stability properties of the generated heteroclinic objects.\",\"PeriodicalId\":340975,\"journal\":{\"name\":\"Chaos: An Interdisciplinary Journal of Nonlinear Science\",\"volume\":\"16 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-03-31\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Chaos: An Interdisciplinary Journal of Nonlinear Science\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1063/5.0156192\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Chaos: An Interdisciplinary Journal of Nonlinear Science","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1063/5.0156192","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Arbitrarily large heteroclinic networks in fixed low-dimensional state space
We consider heteroclinic networks between n∈N nodes where the only connections are those linking each node to its two subsequent neighboring ones. Using a construction method where all nodes are placed in a single one-dimensional space and the connections lie in coordinate planes, we show that it is possible to robustly realize these networks in R6 for any number of nodes n using a polynomial vector field. This bound on the space dimension (while the number of nodes in the network goes to ∞) is a novel phenomenon and a step toward more efficient realization methods for given connection structures in terms of the required number of space dimensions. We briefly discuss some stability properties of the generated heteroclinic objects.