---

Journal of Applied Nonlinear Dynamics 11(4) (2022) 777--788 | DOI:10.5890/JAND.2022.12.001

U.U. Jamilov$^{1,2,3}$, Kh. O. Khudoyberdiev$^{1}$

$^{1}$ V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 9, University str., Tashkent, Uzbekistan

$^{2}$ Akfa University, 264, National Park str., Qibray district, 111221, Tashkent region, Uzbekistan

$^{3}$ National University of Uzbekistan, 4, University str., Tashkent, Uzbekistan

Download Full Text PDF

 

Abstract

In the present paper we consider a family of non-Volterra quadratic stochastic operators depending on the parameters $\alpha, \beta$ and study their trajectory behaviors. We find all fixed and periodic points for an $(\alpha,\beta)$- quadratic stochastic operator on the two-dimensional simplex. A complete description of the set of limit points is given, and we show that such operators have the ergodic property.

References

1. [1]	Lyubich, Y.I. (1992), Mathematical structures in population genetics, vol. 22 of Biomathematics, Springer- Verlag, Berlin.

2. [2]	Bernstein, S. (1942), Solution of a mathematical problem connected with the theory of heredity, The Annals of Mathematical Statistics, 13(1), 53-61.

3. [3]	Ganikhodjaev, N.N., Ganikhodjaev, R.N., and Jamilov(Zhamilov), U.U. (2015), Quadratic stochastic operators and zero-sum game dynamics, Ergod. Th. and Dynam. Sys., 35(5), 1443-1473.

4. [4]	Ganikhodjaev, N.N. Jamilov, U.U., and Mukhitdinov, R.T. (2013), On non - ergodic transformations on $S^3$, Jour. Phys. Conf. Ser., 435, 012005.

5. [5]	Ganikhodjaev, N.N., Jamilov, U.U., and Mukhitdinov, R.T. (2014), Nonergodic quadratic operator for a two-sex population, Ukrainian Math. J., 65(8), 1282-1291.

6. [6]	Ganikhodzhaev, R.N. (1993), Quadratic stochastic operators, Lyapunov functions and tournaments, Sb. Math., 76(2), 489-506.

7. [7]	Ganikhodzhaev, R., Mukhamedov, F., and Rozikov, U. (2011), Quadratic stochastic operators and processes: results and open problems, Infin. Dimens. Anal. Quan. Probab. Relat. Top., 14(2), 279-335.

8. [8]	Jamilov, U.U. (2013), Quadratic stochastic operators corresponding to graphs, Lobachevskii J. Math., 34(2), 148-151.

9. [9]	Jamilov, U.U. (2016), On a family of strictly non-volterra quadratic stochastic operators, Jour. Phys. Conf. Ser., 697, 012013.

10. [10]	Jamilov, U.U. (2016), On symmetric strictly non-Volterra quadratic stochastic operators, Disc. Nonlin. Comp., 5(3), 263-283.

11. [11]	Jamilov, U.U. and Ladra, M. (2016), Non-ergodicity of uniform quadratic stochastic operators, Qual. Theory Dyn. Syst., 15(1), 257-271.

12. [12]	Jamilov, U.U., Ladra, M., and Mukhitdinov, R.T. (2017), On the equiprobable strictly non-Volterra quadratic stochastic operators, Qual. Theory Dyn. Syst., 16(3), 645-655.

13. [13]	Jamilov, U.U., Scheutzow, M., and Wilke-Berenguer, M. (2017), On the random dynamics of Volterra quadratic operators, Ergodic Th. Dynam. Sys., 37(1), 228-243.

14. [14]	Kesten, H. (1970), Quadratic transformations: A model for population growth. I, Advances in Appl. Probability, 2, 1-82.

15. [15]	Khamraev, A.Yu. (2019), On the dynamics of a quasi strictly non-Volterra quadratic stochastic operator, Ukrainian Math. J., 71(8), 1116-1122.

16. [16]	Rozikov, U.A. and Zhamilov, U.U. (2008), F-quadratic stochastic operators, Math. Notes, 83(3-4), 554-559.

17. [17]	Rozikov, U.A. and Zhamilov, U.U. (2011), Volterra quadratic stochastic operators of a two-sex population, Ukrainian Math. J., 63(7), 1136-1153.

18. [18]	Ulam, S.M. (1960), A collection of mathematical problems, Interscience Tracts in Pure and Applied Mathematics, 8, Interscience Publishers, New York-London.

19. [19]	Zakharevich, M.I. (1978), On the behaviour of trajectories and the ergodic hypothesis for quadratic mappings of a simplex, Russ. Math. Surv., 33(6), 265-266.

20. [20]	Zhamilov, U.U. and Rozikov, U.A. (2009), On the dynamics of strictly non-Volterra quadratic stochastic operators on a two-dimensional simplex, Sb. Math., 200(9), 1339-1351.

21. [21]	Mukhamedov, F. and Ganikhodjaev, N. (2015), Quantum Quadratic Operators and Processes, Springer, Berlin.

22. [22]	Lyubich, Y.I. (1974), Iterations of quadratic maps, Math. Econ. Func. Ana. Nauka: Moscow, 109-138. (Russian).

23. [23]	Kolokoltsov, V.N. (2010), Nonlinear Markov Processes and Kinetic Equations, Cambridge Tracts in Math. vol. 182, Cambridge Univ. Press, Cambridge.

24. [24]	Devaney, R.L. (2003), An introduction to chaotic dynamical systems, Studies in Nonlinearity, Westview Press, Boulder, CO.

25. [25]	Ganikhodjaev, N.N., Saburov, M., and Nawi, A.M. (2014), Mutation and chaos in nonlinear models of heredity, The Sci. World Jour., 2014, 835069.

26. [26]	Hardin, A.J.M. and Rozikov, U.A. (2019), A quasi-strictly non-Volterra quadratic stochastic operator, Qual. Theory Dyn. Syst., 18(3), 1013-1029.

27. [27]	Jamilov, U.U., Khudoyberdiev, Kh.O., and Ladra, M. (2020), Quadratic operators corresponding to permutations, Stoch. Anal. Appl., 38(5), 929-938.