Advertisement

On the Dynamics of a Quasistrictly Non-Volterra Quadratic Stochastic Operator

  • 2 Accesses

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.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 199

This is the net price. Taxes to be calculated in checkout.

References

  1. 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. 2.

    R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Westview Press (2003).

  3. 3.

    U. U. Zhalimov and U. A. Rozikov, “On the dynamics of strictly non-Volterra quadratic stochastic operators on a two-dimensional simplex,” Mat. Sb., 200, No. 9, 81–94 (2009).

  4. 4.

    U. U. Jamilov, “On symmetric strictly non-Volterra quadratic stochastic operators,” Discontinuity, Nonlinearity, Complexity, 5, No. 3, 263–283 (2016).

  5. 5.

    A. J. M. Hardin and U. A. Rozikov, A Quasi-Strictly Non-Volterra Quadratic Stochastic Operator, Preprint arXiv: 1808.00229 (2018).

  6. 6.

    H. Kesten, “Quadratic transformations: a model for population growth. I,” Adv. Appl. Probab., 2, No. 1, 1–82 (1970).

  7. 7.

    R. N. Ganikhodzhaev, F. M. Mukhamedov, and U. A. Rozikov, “Quadratic stochastic operators: results and open problems,” Infin. Dimens. Anal. Quantum Probab. Relat. Fields, 14, No. 2, 279–335 (2011).

  8. 8.

    R. N. Ganikhodzhaev, “Quadratic stochastic operators, Lyapunov function, and tournaments,” Mat. Sb., 83, No. 8, 119–140 (1992).

  9. 9.

    R. N. Ganikhodzhaev, “Map of fixed points and Lyapunov functions for one class of discrete dynamical systems,” Mat. Zametki, 56, No. 5, 40–49 (1994).

  10. 10.

    R. N. Ganikhodzhaev, “One family of quadratic stochastic operators acting in S 2,Dokl. Akad. Nauk Uzb. SSR, No. 1, 3–5 (1989).

  11. 11.

    Yu. I. Lubich, Mathematical Structures in Population Genetics [in Russian], Naukova Dumka, Kiev (1983).

  12. 12.

    F. Mukhamedov and N. Ganikhodjaev, Quantum Quadratic Operators and Processes, Springer, New York (2015).

  13. 13.

    F. M. Mukhamedov, “On the infinite-dimensional Volterra quadratic operators,” Usp. Mat. Nauk, 55, No. 6, 149–150 (2000).

  14. 14.

    S. M. Ulam, A Collection of Mathematical Problems, Interscience Publishers, New York (1960).

Download references

Author information

Correspondence to A. Yu. Khamrayev.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 8, pp. 1116–1122, August, 2019.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Khamrayev, A.Y. On the Dynamics of a Quasistrictly Non-Volterra Quadratic Stochastic Operator. Ukr Math J 71, 1273–1281 (2020) doi:10.1007/s11253-019-01712-w

Download citation