On the dynamics of a quasistrictly non-Volterra quadratic stochastic operator. (English. Russian original) Zbl 07622915
Ukr. Math. J. 71, No. 8, 1273-1281 (2020); translation from Ukr. Mat. Zh. 71, No. 8, 1116-1122 (2019).
Summary: We find all fixed and periodic points for a quasistrictly non-Volterra quadratic stochastic operator on a two-dimensional simplex. The description of the limit set of trajectories is presented for this operator.
References:
| [1] | S. N. Bernstein, “Solution of one mathematical problem connected with the theory of hereditary,” Uchen. Zap. Nauch.-Issled. Kafed. Ukr. Otdel. Mat., No. 1, 83-115 (1924). |
| [2] | R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Westview Press (2003). · Zbl 1025.37001 |
| [3] | Zhalimov, Uu; Rozikov, Ua, On the dynamics of strictly non-Volterra quadratic stochastic operators on a two-dimensional simplex, Mat. Sb., 200, 9, 81-94 (2009) · Zbl 1194.47077 · doi:10.4213/sm3962 |
| [4] | Jamilov, Uu, On symmetric strictly non-Volterra quadratic stochastic operators, Discontinuity, Nonlinearity, Complexity, 5, 3, 263-283 (2016) · Zbl 1349.92123 · doi:10.5890/DNC.2016.09.006 |
| [5] | A. J. M. Hardin and U. A. Rozikov, A Quasi-Strictly Non-Volterra Quadratic Stochastic Operator, Preprint arXiv: 1808.00229 (2018). · Zbl 1434.37033 |
| [6] | Kesten, H., Quadratic transformations: a model for population growth. I, Adv. Appl. Probab., 2, 1, 1-82 (1970) · Zbl 0328.92011 · doi:10.2307/3518344 |
| [7] | Ganikhodzhaev, Rn; Mukhamedov, Fm; Rozikov, Ua, Quadratic stochastic operators: results and open problems, Infin. Dimens. Anal. Quantum Probab. Relat. Fields, 14, 2, 279-335 (2011) · Zbl 1242.60067 · doi:10.1142/S0219025711004365 |
| [8] | Ganikhodzhaev, Rn, Quadratic stochastic operators, Lyapunov function, and tournaments, Mat. Sb., 83, 8, 119-140 (1992) · Zbl 0766.47037 |
| [9] | Ganikhodzhaev, Rn, Map of fixed points and Lyapunov functions for one class of discrete dynamical systems, Mat. Zametki, 56, 5, 40-49 (1994) · Zbl 0838.93062 |
| [10] | R. N. Ganikhodzhaev, “One family of quadratic stochastic operators acting in S^2,” Dokl. Akad. Nauk Uzb. SSR, No. 1, 3-5 (1989). · Zbl 0675.47038 |
| [11] | Lubich, Yi, Mathematical Structures in Population Genetics [in Russian] (1983), Kiev: Naukova Dumka, Kiev · Zbl 0593.92011 |
| [12] | Mukhamedov, F.; Ganikhodjaev, N., Quantum Quadratic Operators and Processes (2015), New York: Springer, New York · Zbl 1335.60002 |
| [13] | Mukhamedov, Fm, On the infinite-dimensional Volterra quadratic operators, Usp. Mat. Nauk, 55, 6, 149-150 (2000) · Zbl 1010.60061 · doi:10.4213/rm349 |
| [14] | Ulam, Sm, A Collection of Mathematical Problems (1960), New York: Interscience Publishers, New York · Zbl 0086.24101 |
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