Limit behavior of the trajectories of extreme doubly stochastic quadratic operators on two dimensional simplex

Multi agent systems and consensus problems represents the theoretical aspect of Quadratic Stochastic Operators (QSO). The doubly stochastic quadratic operators (DSQOs) on two-dimensional simplex (2DS) expose a complex problem within QSO and majorization theories in non-linear models. In such models,...

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Main Authors:	Abdulghafor, Rawad, Wani, Sharyar, Abubakar, Adamu, , Shikhah
Format:	Article
Language:	English English
Published:	2019
Subjects:	QA297 Numerical Analysis TA168 Systems engineering
Online Access:	http://irep.iium.edu.my/76300/ http://irep.iium.edu.my/76300/1/Limit%20of%20EDSQO%20on%202DS%20-IJRTE%20-%201.pdf http://irep.iium.edu.my/76300/7/76300-accepted%20letter%20.pdf

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spelling	iium-763002019-12-18T01:39:57Z http://irep.iium.edu.my/76300/ Limit behavior of the trajectories of extreme doubly stochastic quadratic operators on two dimensional simplex Abdulghafor, Rawad Wani, Sharyar Abubakar, Adamu , Shikhah QA297 Numerical Analysis TA168 Systems engineering Multi agent systems and consensus problems represents the theoretical aspect of Quadratic Stochastic Operators (QSO). The doubly stochastic quadratic operators (DSQOs) on two-dimensional simplex (2DS) expose a complex problem within QSO and majorization theories in non-linear models. In such models, the DSQO is considered as a very large class and it is more convenient to study from the sub-classes perspective. Consequently, the DSQO is further classified into the sub-class of extreme doubly stochastic quadratic operators (EDSQOs). In this work, the limit behavior of the trajectories of EDSQO on a two-dimensional simplex (2DS) are studied. In turn, we present further results on the related generalizations of DSQO based on their EDSQO sub-classes. It is demonstrated in the work at hand that, the EDSQO has convergent, fixed, or periodic points. If each positive point of the EDSQO has an undirected interaction (its update is shared among other points), the operator is then convergent. If it has full-directed interaction (its update is not shared among other points), the point is then fixed. However, if two points have full-directed interaction with each other, the points are then considered as periodic. Any convergent operator of EDSQO has a unique fixed point from the positive initial states. Operators that have fixed and periodic points have infinitely fixed points. The work then presents the results of the simulation study on the related behavior of trajectories. Multi agent systems and consensus problems represents the theoretical aspect of Quadratic Stochastic Operators (QSO). The doubly stochastic quadratic operators (DSQOs) on two-dimensional simplex (2DS) expose a complex problem within QSO and majorization theories in non-linear models. In such models, the DSQO is considered as a very large class and it is more convenient to study from the sub-classes perspective. Consequently, the DSQO is further classified into the sub-class of extreme doubly stochastic quadratic operators (EDSQOs). In this work, the limit behavior of the trajectories of EDSQO on a two-dimensional simplex (2DS) are studied. In turn, we present further results on the related generalizations of DSQO based on their EDSQO sub-classes. It is demonstrated in the work at hand that, the EDSQO has convergent, fixed, or periodic points. If each positive point of the EDSQO has an undirected interaction (its update is shared among other points), the operator is then convergent. If it has full-directed interaction (its update is not shared among other points), the point is then fixed. However, if two points have full-directed interaction with each other, the points are then considered as periodic. Any convergent operator of EDSQO has a unique fixed point from the positive initial states. Operators that have fixed and periodic points have infinitely fixed points. The work then presents the results of the simulation study on the related behavior of trajectories. 2019 Article PeerReviewed application/pdf en http://irep.iium.edu.my/76300/1/Limit%20of%20EDSQO%20on%202DS%20-IJRTE%20-%201.pdf application/pdf en http://irep.iium.edu.my/76300/7/76300-accepted%20letter%20.pdf Abdulghafor, Rawad and Wani, Sharyar and Abubakar, Adamu and , Shikhah (2019) Limit behavior of the trajectories of extreme doubly stochastic quadratic operators on two dimensional simplex. International Journal of Recent Technology and Engineering (IJRTE). E-ISSN 2277-3878 (In Press)
repository_type	Digital Repository
institution_category	Local University
institution	International Islamic University Malaysia
building	IIUM Repository
collection	Online Access
language	English English
topic	QA297 Numerical Analysis TA168 Systems engineering
spellingShingle	QA297 Numerical Analysis TA168 Systems engineering Abdulghafor, Rawad Wani, Sharyar Abubakar, Adamu , Shikhah Limit behavior of the trajectories of extreme doubly stochastic quadratic operators on two dimensional simplex
description	Multi agent systems and consensus problems represents the theoretical aspect of Quadratic Stochastic Operators (QSO). The doubly stochastic quadratic operators (DSQOs) on two-dimensional simplex (2DS) expose a complex problem within QSO and majorization theories in non-linear models. In such models, the DSQO is considered as a very large class and it is more convenient to study from the sub-classes perspective. Consequently, the DSQO is further classified into the sub-class of extreme doubly stochastic quadratic operators (EDSQOs). In this work, the limit behavior of the trajectories of EDSQO on a two-dimensional simplex (2DS) are studied. In turn, we present further results on the related generalizations of DSQO based on their EDSQO sub-classes. It is demonstrated in the work at hand that, the EDSQO has convergent, fixed, or periodic points. If each positive point of the EDSQO has an undirected interaction (its update is shared among other points), the operator is then convergent. If it has full-directed interaction (its update is not shared among other points), the point is then fixed. However, if two points have full-directed interaction with each other, the points are then considered as periodic. Any convergent operator of EDSQO has a unique fixed point from the positive initial states. Operators that have fixed and periodic points have infinitely fixed points. The work then presents the results of the simulation study on the related behavior of trajectories. Multi agent systems and consensus problems represents the theoretical aspect of Quadratic Stochastic Operators (QSO). The doubly stochastic quadratic operators (DSQOs) on two-dimensional simplex (2DS) expose a complex problem within QSO and majorization theories in non-linear models. In such models, the DSQO is considered as a very large class and it is more convenient to study from the sub-classes perspective. Consequently, the DSQO is further classified into the sub-class of extreme doubly stochastic quadratic operators (EDSQOs). In this work, the limit behavior of the trajectories of EDSQO on a two-dimensional simplex (2DS) are studied. In turn, we present further results on the related generalizations of DSQO based on their EDSQO sub-classes. It is demonstrated in the work at hand that, the EDSQO has convergent, fixed, or periodic points. If each positive point of the EDSQO has an undirected interaction (its update is shared among other points), the operator is then convergent. If it has full-directed interaction (its update is not shared among other points), the point is then fixed. However, if two points have full-directed interaction with each other, the points are then considered as periodic. Any convergent operator of EDSQO has a unique fixed point from the positive initial states. Operators that have fixed and periodic points have infinitely fixed points. The work then presents the results of the simulation study on the related behavior of trajectories.
format	Article
author	Abdulghafor, Rawad Wani, Sharyar Abubakar, Adamu , Shikhah
author_facet	Abdulghafor, Rawad Wani, Sharyar Abubakar, Adamu , Shikhah
author_sort	Abdulghafor, Rawad
title	Limit behavior of the trajectories of extreme doubly stochastic quadratic operators on two dimensional simplex
title_short	Limit behavior of the trajectories of extreme doubly stochastic quadratic operators on two dimensional simplex
title_full	Limit behavior of the trajectories of extreme doubly stochastic quadratic operators on two dimensional simplex
title_fullStr	Limit behavior of the trajectories of extreme doubly stochastic quadratic operators on two dimensional simplex
title_full_unstemmed	Limit behavior of the trajectories of extreme doubly stochastic quadratic operators on two dimensional simplex
title_sort	limit behavior of the trajectories of extreme doubly stochastic quadratic operators on two dimensional simplex
publishDate	2019
url	http://irep.iium.edu.my/76300/ http://irep.iium.edu.my/76300/1/Limit%20of%20EDSQO%20on%202DS%20-IJRTE%20-%201.pdf http://irep.iium.edu.my/76300/7/76300-accepted%20letter%20.pdf
first_indexed	2023-09-18T21:47:51Z
last_indexed	2023-09-18T21:47:51Z
_version_	1777413569944485888

Limit behavior of the trajectories of extreme doubly stochastic quadratic operators on two dimensional simplex

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