On Non-ergodic Volterra Cubic Stochastic Operators

Farrukh Mukhamedov; Chin Hee Pah; Azizi Rosli

doi:10.1007/s12346-019-00334-8

On Non-ergodic Volterra Cubic Stochastic Operators

Farrukh Mukhamedov, Chin Hee Pah, Azizi Rosli

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

14 Citations (Scopus)

Abstract

Let S^m ^- ¹ be the simplex in R^m, and V: S^m ^- ¹→ S^m ^- ¹ be a nonlinear mapping then this operator satisfies an ergodic theorem if the limit limn→∞1n∑k=1nVk(x)exists for every x∈ S^m ^- ¹. It is a well known fact that this ergodicity may fail for Volterra quadratic operators, so it is natural to characterize all non-ergodic operators. However, there is an ongoing problem even in the low dimensional simplexes. In this paper, we solve the mentioned problem within Volterra cubic stochastic operators acting on two-dimensional simplex.

Original language	English
Pages (from-to)	1225-1235
Number of pages	11
Journal	Qualitative Theory of Dynamical Systems
Volume	18
Issue number	3
DOIs	https://doi.org/10.1007/s12346-019-00334-8
Publication status	Published - Dec 1 2019

Keywords

Cubic stochastic operator
Dynamics
Non-ergodic
Volterra operator

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics
Applied Mathematics

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10.1007/s12346-019-00334-8

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abstract = "Let Sm - 1 be the simplex in Rm, and V: Sm - 1→ Sm - 1 be a nonlinear mapping then this operator satisfies an ergodic theorem if the limit limn→∞1n∑k=1nVk(x)exists for every x∈ Sm - 1. It is a well known fact that this ergodicity may fail for Volterra quadratic operators, so it is natural to characterize all non-ergodic operators. However, there is an ongoing problem even in the low dimensional simplexes. In this paper, we solve the mentioned problem within Volterra cubic stochastic operators acting on two-dimensional simplex.",

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author = "Farrukh Mukhamedov and Pah, {Chin Hee} and Azizi Rosli",

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AU - Mukhamedov, Farrukh

AU - Pah, Chin Hee

AU - Rosli, Azizi

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N2 - Let Sm - 1 be the simplex in Rm, and V: Sm - 1→ Sm - 1 be a nonlinear mapping then this operator satisfies an ergodic theorem if the limit limn→∞1n∑k=1nVk(x)exists for every x∈ Sm - 1. It is a well known fact that this ergodicity may fail for Volterra quadratic operators, so it is natural to characterize all non-ergodic operators. However, there is an ongoing problem even in the low dimensional simplexes. In this paper, we solve the mentioned problem within Volterra cubic stochastic operators acting on two-dimensional simplex.

AB - Let Sm - 1 be the simplex in Rm, and V: Sm - 1→ Sm - 1 be a nonlinear mapping then this operator satisfies an ergodic theorem if the limit limn→∞1n∑k=1nVk(x)exists for every x∈ Sm - 1. It is a well known fact that this ergodicity may fail for Volterra quadratic operators, so it is natural to characterize all non-ergodic operators. However, there is an ongoing problem even in the low dimensional simplexes. In this paper, we solve the mentioned problem within Volterra cubic stochastic operators acting on two-dimensional simplex.

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KW - Dynamics

KW - Non-ergodic

KW - Volterra operator

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On Non-ergodic Volterra Cubic Stochastic Operators

Abstract

Keywords

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