EDSQ Operator on 2DS and Limit Behavior

: This paper evaluates the limit behavior for symmetry interactions networks of set points for nonlinear mathematical models. Nonlinear mathematical models are being increasingly applied to most software and engineering machines. That is because the nonlinear mathematical models have proven to be more e ﬃ cient in processing and producing results. The greatest challenge facing researchers is to build a new nonlinear model that can be applied to di ﬀ erent applications. Quadratic stochastic operators (QSO) constitute such a model that has become the focus of interest and is expected to be applicable in many biological and technical applications. In fact, several QSO classes have been investigated based on certain conditions that can also be applied in other applications such as the Extreme Doubly Stochastic Quadratic Operator (EDSQO). This paper studies the behavior limitations of the existing 222 EDSQ operators on two-dimensional simplex (2DS). The created simulation graph shows the limit behavior for each operator. This limit behavior on 2DS can be classiﬁed into convergent, periodic, and ﬁxed.


Introduction
Discrete dynamical systems have been a subject of interest in many scientific and engineering research applications. The theoretical framework of such systems has been presented in [1][2][3][4][5] using stochastic analysis for population genetics. These dynamical systems consist of set point networks that require symmetrical interactions [6,7]. The work has then been extended in [8] by deploying the majorization technique to derive new notions and properties of the related systems.
A specific case based on linear dynamic systems has been used in [9] to address the agreement issue in the technology science and engineering areas, and has subsequently been reformulated to a nonlinear dynamic system for the same problem [10,11]. In effect, the focus on applications-related studies has shifted to nonlinear dynamical systems in order to address such potential problems. The main Definition 2: The set of s i falls within simplex if s i > 0. The set consists of {0, . . . , 1, . . . , 0} and is called a set of extreme points, while a simplex center set consists of 1 m , 1 m , 1 m .

Definition 3:
The nonlinear model of QSO is defined as [32]: (Vs) k = m i,j,k=1 P ij,k s i s j , where p ij,k is a transaction matrix under conditions [22,32]: The QSO has applications in population genetics, where the population is defined as m species. Here, s 0 = s 0 1 , s 0 2 , . . . , s 0 m is defined as independent members of the population in their initial states. The communication from member i th to member j th called 'parents' allows the transition to produce a new member k. This communication among the members configures a transition matrix P ij,k and produces an inherited factor. Each individual member has a separate transition matrix eventually consisting of the total of these separated matrices with the sum of all its elements being equal to 1. This means that the transition matrix is denoted by P ij,k which has distributed to m matrices P ij,k = p ij,1 p ij,2 . . . p ij,m , where p ij,m is symmetric and nonnegative.
In this case, a steady state s = (s 1 , s 2 , . . . , s m ) of the potential parents xx appears s i s j [32]. Vs ≺ s for all s ∈ S m−1 The set of elements (s [1] , . . . , s [m] ) in DSQO are re-arranged in a non-increasing order s [1] ≥ . . . ≥ s [m] , then: From this point, the majorization concept stipulates that y majorizes s, which is denoted by y s or s is majorized by y. The doubly stochastic matrix P has been defined according to [31] stating that if y = Ps then y s. It follows that in DSQO all s elements belong to the simplex Ps s if P is a doubly stochastic matrix. Therefore, the particular operator of V(s) is called DSQO if the transition matrix P is doubly stochastic in accordance with the majorization theory [31].
In other words, the operator of V(s) is defined a DSQO if the transition matrix P ij,k in Equation (2) satisfies the condition V(s) ≺ s, which means: where α is a subblock in the matrix that contains elements (m − 1) and its sum should be ≤ (m − 1) [22].
In a non-increasing re-arrangement for s m , we obtain: Therefore, the maximum of the set points increases, while the minimum of the set points decreases monotonically through the series of V (n) (s m ) and becomes fixed.
This means that lim n→∞ m i=1 V (n) (s m ) exists.
We can thus conclude that the limit of EDSQO on 2DS exists.

Discussion and Numerical Solution
In this section, we present the simulation results for all operators of EDSQO on 2DS using the MATLAB software.
First, we outline all 37 EDSQOs on 2DS as follows: V(s 1 ) = s 2 2 + s 3 2 + s 1 s 2 V(s 2 ) = s 2 s 3 + 2s 1  where V 1 to V 37 are EDSQOs names, and each of the 37 EDSQOs will have 3! permuted operators. For instance, for operator V 1 ,                                         In this section, we explain the Figures 1-13. The initial statuses (s 1 , s 2 , s 3 ) are random, I is the number of iterations for convergence, and T is the computation time to converge. Each figure is assigned a name under EDSQO, followed by the number of operators from V 1 to V 37 , where each operator from V 1 to V 37 is given one row of Figures (a, b, c, d, e, f ) which are the permutations of 3!.
From the simulation result it can be observed that are altogether 222 EDSQOs, whereby 198 EDSQOs always converge to the average 1 3 or the center (see Figures 1-13). In the case of 18 EDSQOs found that each has one fixed point and two converged points Figures 2-4, Figures 7-10, Figure 12). Moreover, five EDSQOs (V 28a , V 28b , V 28c , V 28d , V 28e ) had periodic points (see Figure 10), and one EDSQO had only fixed points (see Figure 10). This simulation analysis proves that the limit behavior of DSQOs in the form of converged or periodic or fixed points. That is due to the fact that if the EDSQO has positive points and none of them has a selfish interaction, these points converge (the 198 EDSQOs). However, if one of these points has a selfish interaction (the 18 EDSQOs and V 28 f ), they do not change and are fixed. Consequently, if there are two or more points that have selfish communications, they are periodized (the five EDSQOs).
In conclusion, the study of the limit behavior of all DSQOs on finite dimensional simplex is considerably complex. This is due to the fact that the set points of DSQOs represent a relatively large class. Thus, we have taken a subclass of DSQO that can be studied more conveniently and be used to create a general theory for the entire DSQO class. Therefore, the study has been confined to the EDSQOs class on 2DS due to its low-complexity computation. The limit behavior of EDSQOs on 2DS has been determined and the analysis result has been generalized to the DSQO class.

Conclusion and Future Work
We have attempted to analyze a considerably challenging issue related to the nonlinear mathematical model. We have deliberated the trajectories limit behavior of all extreme points for DSQO on 2DS. Subsequently, we have generalized the implications of the result of those EDSQO trajectories' limit behavior by applying them to the whole class of DSQO. In the process, our work has demonstrated the existence of the EDSQOs' limit behavior. Through simulation, we have been able to observe the set points' convergence to the center if they have positive statuses and unselfish communications or their fixation if they have positive statuses and selfish communications or their periodization if they have two or more points with selfish communications. For any future work on this subject, we recommend the study of EDSQ operators on infinite dimensional simplex. Furthermore, it is recommended to evaluate the model for certain specific applications and compare it with existing similar models for the consensus problem.

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A. The EDSQOs on 2DS
The permutation probabilities distribution 3! for each distribution of the 37 matrices P ij,k i and operators V i of EDSQOs on 2DS are 222 where i the name of matrices and operators from 1 to 37, while a, b, c, d, e, f are the factorial of 3 for each matrix and operator as follows: The permutation probabilities distribution 3! for each distribution of the 37 matrices , and operators of EDSQOs on 2DS are 222 where the name of matrices and operators from 1 to 37, while , , , , , are the factorial of 3 for each matrix and operator as follows: