# Dynamics of doubly stochastic quadratic operators on a finite-dimensional simplex

Open Mathematics (2016)

• Volume: 14, Issue: 1, page 509-519
• ISSN: 2391-5455

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## Abstract

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The present paper focuses on the dynamics of doubly stochastic quadratic operators (d.s.q.o) on a finite-dimensional simplex. We prove that if a d.s.q.o. has no periodic points then the trajectory of any initial point inside the simplex is convergent. We show that if d.s.q.o. is not a permutation then it has no periodic points on the interior of the two dimensional (2D) simplex. We also show that this property fails in higher dimensions. In addition, the paper also discusses the dynamics classifications of extreme points of d.s.q.o. on two dimensional simplex. As such, we provide some examples of d.s.q.o. which has a property that the trajectory of any initial point tends to the center of the simplex. We also provide and example of d.s.q.o. that has infinitely many fixed points and has infinitely many invariant curves. We therefore came-up with a number of evidences. Finally, we classify the dynamics of extreme points of d.s.q.o. on 2D simplex.

## How to cite

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Rawad Abdulghafor, et al. "Dynamics of doubly stochastic quadratic operators on a finite-dimensional simplex." Open Mathematics 14.1 (2016): 509-519. <http://eudml.org/doc/285534>.

abstract = {The present paper focuses on the dynamics of doubly stochastic quadratic operators (d.s.q.o) on a finite-dimensional simplex. We prove that if a d.s.q.o. has no periodic points then the trajectory of any initial point inside the simplex is convergent. We show that if d.s.q.o. is not a permutation then it has no periodic points on the interior of the two dimensional (2D) simplex. We also show that this property fails in higher dimensions. In addition, the paper also discusses the dynamics classifications of extreme points of d.s.q.o. on two dimensional simplex. As such, we provide some examples of d.s.q.o. which has a property that the trajectory of any initial point tends to the center of the simplex. We also provide and example of d.s.q.o. that has infinitely many fixed points and has infinitely many invariant curves. We therefore came-up with a number of evidences. Finally, we classify the dynamics of extreme points of d.s.q.o. on 2D simplex.},
author = {Rawad Abdulghafor, Farruh Shahidi, Akram Zeki, Sherzod Turaev},
journal = {Open Mathematics},
keywords = {Doubly stochastic quadratic operators; Fixed point; Trajectory; Extreme point; Simplex; doubly stochastic quadratic operators; fixed point; trajectory; extreme point; simplex},
language = {eng},
number = {1},
pages = {509-519},
title = {Dynamics of doubly stochastic quadratic operators on a finite-dimensional simplex},
url = {http://eudml.org/doc/285534},
volume = {14},
year = {2016},
}

TY - JOUR
AU - Farruh Shahidi
AU - Akram Zeki
AU - Sherzod Turaev
TI - Dynamics of doubly stochastic quadratic operators on a finite-dimensional simplex
JO - Open Mathematics
PY - 2016
VL - 14
IS - 1
SP - 509
EP - 519
AB - The present paper focuses on the dynamics of doubly stochastic quadratic operators (d.s.q.o) on a finite-dimensional simplex. We prove that if a d.s.q.o. has no periodic points then the trajectory of any initial point inside the simplex is convergent. We show that if d.s.q.o. is not a permutation then it has no periodic points on the interior of the two dimensional (2D) simplex. We also show that this property fails in higher dimensions. In addition, the paper also discusses the dynamics classifications of extreme points of d.s.q.o. on two dimensional simplex. As such, we provide some examples of d.s.q.o. which has a property that the trajectory of any initial point tends to the center of the simplex. We also provide and example of d.s.q.o. that has infinitely many fixed points and has infinitely many invariant curves. We therefore came-up with a number of evidences. Finally, we classify the dynamics of extreme points of d.s.q.o. on 2D simplex.
LA - eng
KW - Doubly stochastic quadratic operators; Fixed point; Trajectory; Extreme point; Simplex; doubly stochastic quadratic operators; fixed point; trajectory; extreme point; simplex
UR - http://eudml.org/doc/285534
ER -

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