{"title":"The rotation number in one-dimensional maps: definition and applications","authors":"G. Livadiotis, N. Voglis","doi":"10.1088/0305-4470/39/49/011","DOIUrl":null,"url":null,"abstract":"A rotation number in the case of one-dimensional maps is introduced. As is shown, this rotation number is equivalent to the already known rotation number in the case of two-dimensional maps. The definition of the rotation number is given in two steps. First, it is defined for periodic orbits inside a window of organized motion (WOM). We show that in this case our definition coincides with the definition of the over-rotation number. Then, our definition is further generalized for chaotic orbits outside the WOMs. Thus, we obtain a unified definition of the rotation number for the whole area of the chaotic zone of the bifurcation diagram, having a number of useful applications. Namely, it can be used as a tool to distinguish whether an orbit is contained within a WOM or not, as a tool of numerical location of the bifurcation points, of the band mergings, as well as of the boundary points of a WOM. Finally, a method of numerical calculation of the percentage of the cumulative width of the WOMs in every particular segment (chaotic band) of the chaotic zone is given.","PeriodicalId":87442,"journal":{"name":"Journal of physics A: Mathematical and general","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2006-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of physics A: Mathematical and general","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1088/0305-4470/39/49/011","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 4
Abstract
A rotation number in the case of one-dimensional maps is introduced. As is shown, this rotation number is equivalent to the already known rotation number in the case of two-dimensional maps. The definition of the rotation number is given in two steps. First, it is defined for periodic orbits inside a window of organized motion (WOM). We show that in this case our definition coincides with the definition of the over-rotation number. Then, our definition is further generalized for chaotic orbits outside the WOMs. Thus, we obtain a unified definition of the rotation number for the whole area of the chaotic zone of the bifurcation diagram, having a number of useful applications. Namely, it can be used as a tool to distinguish whether an orbit is contained within a WOM or not, as a tool of numerical location of the bifurcation points, of the band mergings, as well as of the boundary points of a WOM. Finally, a method of numerical calculation of the percentage of the cumulative width of the WOMs in every particular segment (chaotic band) of the chaotic zone is given.