# EDSQ Operator on 2DS and Limit Behavior

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## Abstract

**:**

## 1. Introduction

## 2. Methods

**Definition**

**1:**

**Definition**

**2:**

**Definition**

**3:**

**Definition**

**4:**

**Definition**

**5:**

**Definition**

**6:**

- a.
- ${p}_{ij,k}=0or\frac{1}{2}or1;$
- b.
- ${p}_{ii,k}=0or1;$

**Definition**

**7:**

**Definition**

**8:**

**Definition**

**9:**

**Definition**

**10:**

**Definition**

**11:**

## 3. Theoretical Result

**Theorem**

**1:**

**Proof:**

## 4. Discussion and Numerical Solution

## 5. Conclusion and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. The EDSQOs on 2DS

## References

- Lyubich, Y.I.; Vulis, D.; Karpov, A.; Akin, E. Mathematical structures in population genetics. Biomathematics
**1992**, 22, 373. [Google Scholar] - Abdulghafor, R.; Shahidi, F.; Zeki, A.; Turaev, S. Dynamics classifications of extreme doubly stochastic quadratic operators on 2d simplex. In Advanced Computer and Communication Engineering Technology; Springer: Berlin/Heidelberg, Germany, 2016; pp. 323–335. [Google Scholar]
- Abdulghafor, R.; Shahidi, F.; Zeki, A.; Turaev, S. Dynamics of doubly stochastic quadratic operators on a finite-dimensional simplex. Open Math.
**2016**, 14, 509–519. [Google Scholar] [CrossRef] - Shahidi, F.; Ganikhodzhaev, R.; Abdulghafor, R. The dynamics of some extreme doubly stochastic quadratic operators. Middle East J. Sci. Res.
**2013**, 13, 59–63. [Google Scholar] - Abdulghafor, R.; Turaev, S.; Abubakar, A.; Zeki, A. The extreme doubly stochastic quadratic operators on two dimensional simplex. In Proceedings of the 2015 4th International Conference on Advanced Computer Science Applications and Technologies (ACSAT), Kuala Lumpur, Malaysia, 8–10 December 2015; pp. 192–197. [Google Scholar]
- Abdulghafor, R.; Abdullah, S.S.; Turaev, S.; Hassan, R. The nonlinear limit control of EDSQOs on finite dimensional simplex. Automatika
**2019**, 60, 404–412. [Google Scholar] [CrossRef][Green Version] - Abdulghafor, R.; Turaev, S.; Zeki, A. Necessary and Sufficient Conditions for Complementary Stochastic Quadratic Operators of Finite-Dimensional Simplex. Sukkur IBA J. Comput. Math. Sci.
**2017**, 1, 22–27. [Google Scholar] [CrossRef][Green Version] - Ganikhodzhaev, R.N. On the definition of bistochastic quadratic operators. Russ. Math. Surv.
**1993**, 48, 244–246. [Google Scholar] [CrossRef] - DeGroot, M.H. Reaching a consensus. J. Am. Stat. Assoc.
**1974**, 69, 118–121. [Google Scholar] [CrossRef] - Abdulghafor, R.; Turaev, S.; Zeki, A.; Shahidi, F. The convergence consensus of multi-agent systems controlled via doubly stochastic quadratic operators. In Proceedings of the 2015 International symposium on agents, multi-agent systems and robotics (ISAMSR), Putrajaya, Malaysia, 18–19 August 2015; pp. 59–64. [Google Scholar]
- Abdulghafor, R.; Turaev, S.; Tamrin, I. Nonlinear consensus for multi-agent systems using positive intractions of doubly stochastic quadratic operators. Int. J. Perceptive Cogn. Comput.
**2016**, 2, 19–22. [Google Scholar] [CrossRef] - Bernstein, S. Solution of a mathematical problem connected with the theory of heredity. Ann. Math. Stat.
**1942**, 13, 53–61. [Google Scholar] [CrossRef] - Abdulghafor, R.; Almotairi, S.; Almohamedh, H.; Turaev, S.; Almutairi, B. Nonlinear Consensus Protocol Modified from Doubly Stochastic Quadratic Operators in Networks of Dynamic Agents. Symmetry
**2019**, 11, 1519. [Google Scholar] [CrossRef][Green Version] - Abdulghafor, R.; Abdullah, S.S.; Turaev, S.; Zeki, A.; Al-Shaikhli, I. Linear and nonlinear stochastic distribution for consensus problem in multi-agent systems. Neural Comput. Appl.
**2018**, 32, 261–277. [Google Scholar] - Abdulghafor, R.; Abdullah, S.S.; Turaev, S.; Othman, M. An overview of the consensus problem in the control of multi-agent systems. Automatika
**2018**, 59, 143–157. [Google Scholar] [CrossRef] - Abdulghafor, R.; Almotairi, S.; Almohamedh, H.; Almutairi, B.; Bajahzar, A.; Almutairi, S.S. A Nonlinear Convergence Consensus: Extreme Doubly Stochastic Quadratic Operators for Multi-Agent Systems. Symmetry
**2020**, 12, 1519. [Google Scholar] [CrossRef][Green Version] - Abdulghafor, R.; Turaev, S. Consensus of fractional nonlinear dynamics stochastic operators for multi-agent systems. Inf. Fusion
**2018**, 44, 1–21. [Google Scholar] [CrossRef] - Abdulghafor, R.; Turaev, S.; Izzuddin, M. Nonlinear Models for Distributed Consensus Modified from DSQO in Networks of Dynamic Agents. In Proceedings of the 4th International Conference on Mathematical Sciences, Putrajaya, Malaysia, 15–17 November 2016. [Google Scholar]
- Abdulghafor, R.; Turaev, S.; Zeki, A.; Al-Shaikhli, I. Reach a nonlinear consensus for MAS via doubly stochastic quadratic operators. Int. J. Control
**2018**, 91, 1431–1459. [Google Scholar] [CrossRef] - Abdulghafor, R.; Turaev, S.; Zeki, A.; Abubaker, A. Nonlinear convergence algorithm: Structural properties with doubly stochastic quadratic operators for multi-agent systems. J. Artif. Intell. Soft Comput. Res.
**2018**, 8, 49–61. [Google Scholar] [CrossRef][Green Version] - Ganikhodzhaev, R.N. Quadratic stochastic operators, Lyapunov functions, and tournaments. Russ. Acad. Sci. Sb. Math.
**1993**, 76, 489. [Google Scholar] [CrossRef] - Ganikhodzhaev, R.; Shahidi, F. Doubly stochastic quadratic operators and Birkhoff’s problem. Linear Algebra Appl.
**2010**, 432, 24–35. [Google Scholar] [CrossRef][Green Version] - Shahidi, F.A.; Osman, M.T.A. The Limit behavior of the trajectories of dissipative quadratic stochastic operators on finite-dimensional simplex. J. Differ. Equations Appl.
**2013**, 19, 357–371. [Google Scholar] [CrossRef] - Shahidi, F. On dissipative quadratic stochastic operators. Appl. Math. Inf. Sci.
**2008**, 2, 211–223. [Google Scholar] - Shahidi, F. Necessary and sufficient conditions for doubly stochasticity of infinite-dimensional quadratic operators. Linear Algebra Appl.
**2013**, 438, 96–110. [Google Scholar] [CrossRef] - Mirsky, L. Even doubly-stochastic matrices. Math. Ann.
**1961**, 144, 418–421. [Google Scholar] [CrossRef] - Mirsky, L. Results and Problems in the Theory of Doubly-Stochastic Matrices. Group
**1963**, 334, 319–334. [Google Scholar] [CrossRef] - Ryff, J.V. On the representation of doubly stochastic operators. Pacific J. Math
**1963**, 13, 1379–1389. [Google Scholar] [CrossRef] - Ryff, J.V. Orbits of L1 Functions under Doubly Stochastic Transformations. Trans. Am. Math. Soc. JSTOR
**1965**, 117, 92–100. [Google Scholar] - Wang, Z.; Duan, Z.; Cao, J. Impulsive synchronization of coupled dynamical networks with nonidentical Duffing oscillators and coupling delays. Chaos Interdiscip. J. Nonlinear Sci.
**2012**, 22, 13140. [Google Scholar] [CrossRef] - Olkin, I.; Marshall, A.W. Inequalities: Theory of Majorization and Its Applications; Academic Press: Cambridge, MA, USA, 2016; Volume 143. [Google Scholar]
- Ganikhodzhaev, R.N.; Rozikov, U.A. Quadratic stochastic operators: Results and open problems. arXiv
**2009**, arXiv:0902.4207. [Google Scholar] [CrossRef]

**Figure 1.**Limit behavior of permuted operators of V1, V2, and V3 for Extreme Doubly Stochastic Quadratic Operator (EDSQO) on two-dimensional simplex (2DS).

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**MDPI and ACS Style**

Abdulghafor, R.; Almohamedh, H.; Almutairi, B.; Wani, S.; Alharbi, A.; Almutairi, S.S.; Almotairi, S.
EDSQ Operator on 2DS and Limit Behavior. *Symmetry* **2020**, *12*, 820.
https://doi.org/10.3390/sym12050820

**AMA Style**

Abdulghafor R, Almohamedh H, Almutairi B, Wani S, Alharbi A, Almutairi SS, Almotairi S.
EDSQ Operator on 2DS and Limit Behavior. *Symmetry*. 2020; 12(5):820.
https://doi.org/10.3390/sym12050820

**Chicago/Turabian Style**

Abdulghafor, Rawad, Hamad Almohamedh, Badr Almutairi, Sharyar Wani, Abdullah Alharbi, Sulaiman Sulmi Almutairi, and Sultan Almotairi.
2020. "EDSQ Operator on 2DS and Limit Behavior" *Symmetry* 12, no. 5: 820.
https://doi.org/10.3390/sym12050820