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, 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