{"title":"关于Volterra和非Volterra三次随机算子的动力学","authors":"U. Jamilov, A. Y. Khamrayev","doi":"10.1080/14689367.2021.2006150","DOIUrl":null,"url":null,"abstract":"We consider Volterra and non-Volterra cubic stochastic operators. For a Volterra cubic stochastic operator defined on the two-dimensional simplex, it is shown that the vertices and the centre of the simplex are fixed points. The trajectory of such an operator starting from any point from the boundary of the simplex is convergent, and the trajectory of such an operator starting from any point from the interior of the simplex except the centre does not converge. Moreover, therein proven the mean of any trajectory does not converge. For a non-Volterra cubic stochastic operator defined on the two-dimensional simplex, it is proved that the uniqueness of a fixed point, which is repelling and any trajectory starting from the boundary of the simplex converges to a periodic trajectory which consists of three vertices of the simplex. The set of limit points of the trajectory starting from the interior of the simplex except the centre is an infinite subset of the boundary of the simplex.","PeriodicalId":50564,"journal":{"name":"Dynamical Systems-An International Journal","volume":"37 1","pages":"66 - 82"},"PeriodicalIF":0.5000,"publicationDate":"2021-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On dynamics of Volterra and non-Volterra cubic stochastic operators\",\"authors\":\"U. Jamilov, A. Y. Khamrayev\",\"doi\":\"10.1080/14689367.2021.2006150\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider Volterra and non-Volterra cubic stochastic operators. For a Volterra cubic stochastic operator defined on the two-dimensional simplex, it is shown that the vertices and the centre of the simplex are fixed points. The trajectory of such an operator starting from any point from the boundary of the simplex is convergent, and the trajectory of such an operator starting from any point from the interior of the simplex except the centre does not converge. Moreover, therein proven the mean of any trajectory does not converge. For a non-Volterra cubic stochastic operator defined on the two-dimensional simplex, it is proved that the uniqueness of a fixed point, which is repelling and any trajectory starting from the boundary of the simplex converges to a periodic trajectory which consists of three vertices of the simplex. The set of limit points of the trajectory starting from the interior of the simplex except the centre is an infinite subset of the boundary of the simplex.\",\"PeriodicalId\":50564,\"journal\":{\"name\":\"Dynamical Systems-An International Journal\",\"volume\":\"37 1\",\"pages\":\"66 - 82\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2021-11-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Dynamical Systems-An International Journal\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1080/14689367.2021.2006150\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Dynamical Systems-An International Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1080/14689367.2021.2006150","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
On dynamics of Volterra and non-Volterra cubic stochastic operators
We consider Volterra and non-Volterra cubic stochastic operators. For a Volterra cubic stochastic operator defined on the two-dimensional simplex, it is shown that the vertices and the centre of the simplex are fixed points. The trajectory of such an operator starting from any point from the boundary of the simplex is convergent, and the trajectory of such an operator starting from any point from the interior of the simplex except the centre does not converge. Moreover, therein proven the mean of any trajectory does not converge. For a non-Volterra cubic stochastic operator defined on the two-dimensional simplex, it is proved that the uniqueness of a fixed point, which is repelling and any trajectory starting from the boundary of the simplex converges to a periodic trajectory which consists of three vertices of the simplex. The set of limit points of the trajectory starting from the interior of the simplex except the centre is an infinite subset of the boundary of the simplex.
期刊介绍:
Dynamical Systems: An International Journal is a world-leading journal acting as a forum for communication across all branches of modern dynamical systems, and especially as a platform to facilitate interaction between theory and applications. This journal publishes high quality research articles in the theory and applications of dynamical systems, especially (but not exclusively) nonlinear systems. Advances in the following topics are addressed by the journal:
•Differential equations
•Bifurcation theory
•Hamiltonian and Lagrangian dynamics
•Hyperbolic dynamics
•Ergodic theory
•Topological and smooth dynamics
•Random dynamical systems
•Applications in technology, engineering and natural and life sciences