Investigation of the dynamic behaviors of the nonlinear operators generated from ξ (as)-QSO


  • Hamzah Garalleh
  • Ahmad Termimi Ab Ghani
  • I. Qaralleh





Fixed point, limiting point, quadratic stochastic operator.


A quadratic stochastic operator (QSO) exhibits the time development of various species in biology. Several QSOs have been examined by Lotka and Volterra. The main problem in a nonlinear operators is to explore their behavior. The behavior of a nonlinear operators have not been studied in comprehensively even QSOs which are the simplest a nonlinear operators. To address this problem, many classes of QSO were introduced. This paper aims to examine the behavior of six an operators selected from different classes of ξ (as)-QSO.


[1] Bernstein S., Solution of a mathematical problem connected with the theory of heredity. Annals of Math. Statis. 13(1942), 53–61.

[2] Ganikhodjaev N. N., Rozikov U. A., on quadratic stochastic operators generated by Gibbs distributions. Regul. Chaotic Dyn. 11 (2006), 467473.

[3] Ganikhodjaev N.N., An application of the theory of Gibbs distributions to mathematical genetics. Doklady Math 61 (2000), 321–323.

[4] Ganikhodzhaev N. N., Mukhitdinov R. T., on a class of measures corresponding to quadratic operators, Dokl. Akad. Nauk Rep. Uzb. No.

3 (1995), 3–6 (Russian).

[5] Ganikhodzhaev R. N., A family of quadratic stochastic operators that act in S2. Dokl. Akad. Nauk UzSSR. No. 1 (1989), 3–5. (Russian)

[6] Ganikhodzhaev R. N., Quadratic stochastic operators, Lyapunov functions and tournaments.Acad. Sci. Sb. Math. 76 no. 2 (1993), 489-506.

[7] Ganikhodzhaev R. N., Dzhurabaev A. M., The set of equilibrium states of quadratic stochastic operators of type Vp. Uzbek Math. Jour. No. 3 (1998), 23-27. (Russian)

[8] Ganikhodzhaev R. N., Abdirakhmanova R. E., Description of quadratic automorphisms of a finite-dimensional simplex. Uzbek. Math. (2002), 7–16.(Russian).

[9] Hofbauer J. and Sigmund K., The theory of evolution and dynamical systems. Mathematical aspects of selection, Cambridge Univ. Press, 1988.

[10] Kesten H., Quadratic transformations: a model for population growth.I.II, Adv. Appl.Probab, 2 01 (1970), 1–82.

[11] Lyubich Yu. I., Mathematical structures in population genetics, Springer-Verlag, (1992).

[12] Mukhamedov, Farrukh, Izzat Qaralleh, and W. N. F. A. W. Rozali. â€On x a-quadratic stochastic operators on 2D simplex.†Sains Malaysiana43.8 (2014): 1275-1281.

[13] Alrwashdeh, Saad Sabe. â€Assessment of Photovoltaic Energy Production at Different Locations in Jordan.†International Journal of

Renewable Energy Research-IJRER 8.2 (2018).

[14] Rozikov U.A., Zada A. On `- Volterra Quadratic stochastic operators. Inter. Journal Biomath. 3 (2010), 143–159.

[15] Rozikov U.A., Zada A. `-Volterra quadratic stochastic operators: Lyapunov functions, trajectories, Appl. Math. & Infor. Sci. 6 (2012), 329–335


[16] Rozikov U.A., Zhamilov U.U., On F-quadratic stochastic operators. Math. Notes. 83 (2008), 554–559.

[17] Rozikov U.A., Zhamilov U.U. On dynamics of strictly non-Volterra quadratic operators defined on the two dimensional simplex. Sbornik:

Math. 200 no.9 (2009), 81–94.

[18] Stein, P.R. and Ulam, S.M., Non-linear transformation studies on electronic computers, 1962, Los Alamos Scientific Lab., N. Mex.

[19] Alrwashdeh, Saad S.â€Investigation of Wind Energy Production at Different Sites in Jordan Using the Site Effectiveness Method.†Energy Engineering 116.1 (2019): 47-59.

[20] Ulam S.M., Problems in Modern Math., New York; Wiley, 1964.

[21] Mukhamedov Farrukh, Mansour Saburov, and Izzat Qaralleh. â€On x (s)-Quadratic Stochastic Operators on Two-Dimensional Simplex and Their Behavior.†Abstract and Applied Analysis. Vol.2013 (2013), 1–13.

[22] Alsarayreh, A., Qaralleh, I., & Ahmad, M. Z. x (as) -Quadratic Stochastic Operators in Two-Dimensional Simplex and Their Behavior. JP

Journal of Algebra, Number Theory and Applications. 39 5(2017), 737–770.

[23] Hofbauer J., Hutson V. and Jansen W., Coexistence for systems governed by difference equations of Lotka-Volterra type. Jour. Math. Biology, 25 (1987), 553–570.

[24] Ganikhodzhaev R. N., Eshmamatova D. B., Quadratic automorphisms of a simplex and the asymptotic behavior of their trajectories,

Vladikavkaz. Math. Jour. 8 no. 2 (2006), 12–28. (Russian) magnitude. Uzbek. Math. Jour. No. 4 (2000), 16–21.(Russian)

[25] Ganikhodzhaev R., Mukhamedov F., Rozikov U., Quadratic stochastic operators and processes: results and open problems, Infin. Dmens. Anal. Quantum Probab. Relat. Top. 14 (2011), 270–335.

[26] El-Qader, Hamza Abd, Ahmad Termimi Ab Ghani, and Izzat Qaralleh. Classification and study of a new class of x (as)-QSO.†arXiv preprint arXiv: 1807.11210 (2018). 12–28.(Russian).

[27] Alrwashdeh, Saad S.â€Modelling of Operating Conditions of Conduction

Heat Transfer Mode Using Energy 2D Simulation.†International Journal of Online Engineering (iJOE) 14.09 (2018): 200-207.

[28] Ince, Utku U., et al.â€Effects of compression on water distribution in gas diffusion layer materials of PEMFC in a point injection device

By means of synchrotron X-ray imaging.†International Journal of Hydrogen Energy 43.1

[29] Alsarayreh, Abdelwahab, Izzat Qaralleh, Muhammad Zaini Ahmad,Basma Al-Shutnawi, and Saba Al-Kaseasbeh. â€Global and local behavior of a class of x (as)-QSO.â€

[30] Qaralleh, I. (2017). Classification of a new subclass of x (as)-QSO and its dynamics. JOURNAL OF MATHEMATICS AND COMPUTER

SCIENCE-JMCS, 17(4), 535-544.

[31] Mukhamedov, Farrukh, and Nasir Ganikhodjaev. Quantum quadratic operators and processes. Vol. 2133. Berlin: Springer, 2015.

View Full Article: