Geodesic
Dynamics of quadratic Volterra-type stochastic operators corresponding to strange tournaments
Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 1, pp. 85-99 
R. N. Ganikhodzhaev; K. A. Kurganov; M. A. Tadzhieva; F. Kh. Khaidarov. Dynamics of quadratic Volterra-type stochastic operators corresponding to strange tournaments. Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 1, pp. 85-99. http://geodesic.mathdoc.fr/item/VMJ_2024_26_1_a6/
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Voir la notice de l'article provenant de la source Math-Net.Ru

By studying the dynamics of these operators on the simplex, focusing on the presence of an interior fixed point, we investigate the conditions under which the operators exhibit nonergodic behavior. Through rigorous analysis and numerical simulations, we demonstrate that certain parameter regimes lead to nonergodicity, characterized by the convergence of initial distributions to a limited subset of the simplex. Our findings shed light on the intricate dynamics of quadratic stochastic operators with interior fixed points and provide insights into the emergence of nonergodic behavior in complex dynamical systems. Also, the nonergodicity of quadratic stochastic operators of Volterra type with an interior fixed point defined in a simplex introduces additional complexity to the already intricate dynamics of such systems. In this context, the presence of an interior fixed point within the simplex further complicates the exploration of the state space and convergence properties of the operator. In this paper, we give sufficiency and necessary conditions for the existence of strange tournaments. Also, we prove the nonergodicity of quadratic stochastic operators of Volterra type with an interior fixed point, defined in a simplex.

[1] Fisher, M. E. and Goh, B. S., “Stability in a Class of Discrete-Time Models of Interacting Populations”, Journal of Mathematical Biology, 4 (1977), 265–274 | DOI | MR | Zbl

[2] Bernstein, S. N., “Solution of a Mathematical Problem Connected with the Theory of Heredity”, The Annals of Mathematical Statistics, 13:1 (1942), 53–61 | DOI | MR | Zbl

[3] Dohtani, A., “Occurrence of Chaos in Higher-Dimensional Discrete-Time Systems”, SIAM Journal on Applied Mathematics, 52:6 (1992), 1707–1721 | DOI | MR | Zbl

[4] Hofbauer, J. and Sigmund, K., The Theory of Evolution and Dynamical Systems, Cambridge University Press, 1988 | MR | Zbl

[5] Lyubich, Yu. I., Mathematical Structures in Population Genetics, Springer-Verlag, 1992 | MR | Zbl

[6] Ulam, S. M., Problems in Modern Mathematics, John Wiley Sons, New York, 1964 | MR | Zbl

[7] Ganikhodzhaev, N. N. and Mukhamedov, F. M., “On Quantum Quadratic Stochastic Processes and Ergodic Theorems for Such Processes”, Uzbek Mathematical Journal, 3 (1997), 8–20 (in Russian) | MR | Zbl

[8] Mukhamedov, F. M., “On Expansion of Quantum Quadratic Stochastic Processes into Fibrewise Markov Processes Defined on von Neumann Algebras”, Izvestiya: Mathematics, 68:5 (2004), 1009–1024 | DOI | MR | Zbl

[9] Ganikhodzhaev, N. N. and Mukhamedov, F. M., “Ergodic Properties of Discrete Quadratic Stochastic Processes Defined on Von Neumann Algebras”, Izvestiya: Mathematics, 64:5 (2000), 873–890 | DOI | MR | Zbl

[10] Zakharevich, M. I., “The Behavior of Trajectories and the Ergodic Hypothesis for Quadratic Mappings of a Simplex”, Russian Mathematical Surveys, 33:6 (1978), 207–208 | DOI | MR | Zbl

[11] Ganikhodzhaev, N. N. and Zanin, D. V., “On a Necessary Condition for the Ergodicity of Quadratic Operators Defined on a Two-Dimensional Simplex”, Russian Mathematical Surveys, 59:3 (2004), 571–572 | DOI | MR | Zbl

[12] Ganikhodzhaev R. N., “Quadratic Stochastic Operators, Lyapunov Function and Tournaments”, Russian Academy of Sciences. Sbornik. Mathematics, 76:2 (1993), 489–506 | DOI | MR