On a Family of Non-Volterra Quadratic Operators Acting on a Simplex

• Jamilov, Uygun ^[1] ; Ladra, Manuel ^[2]
  1. [1] Institute of Nuclear Physics

     Institute of Nuclear Physics

     Uzbekistán

     
  2. [2] Universidade de Santiago de Compostela

     Universidade de Santiago de Compostela

     Santiago de Compostela, España

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 3, 2020
• Idioma: inglés
• DOI: 10.1007/s12346-020-00433-x
• Enlaces
  • Texto completo
• Resumen

  • In the present paper, we consider a convex combination of non-Volterra quadratic stochastic operators defined on a finite-dimensional simplex depending on a parameter α and study their trajectory behaviours. We showed that for any α∈[0,1) the trajectories of such operator converge to a fixed point. For α=1 any trajectory of the operator converges to a periodic trajectory.

• Referencias bibliográficas
  • 1. Bernstein, S.: The solution of a mathematical problem related to the theory of heredity, Uchn. Zapiski. NI Kaf. Ukr. Otd. Mat. 1, 83–115...
  • 2. Devaney, R. L.: An introduction to chaotic dynamical systems. In: Studies in Nonlinearity. Westview Press, Boulder. Reprint of the second...
  • 3. Ganikhodjaev, N.N., Ganikhodjaev, R.N., Jamilov, U.U.: Quadratic stochastic operators and zero-sum game dynamics. Ergod. Theory Dyn. Syst....
  • 4. Ganikhodjaev, N.N., Saburov, M., Nawi, A.M.: Mutation and chaos in nonlinear models of heredity. Sci. World J. (2014). https://doi.org/10.1155/2014/835069
  • 5. Ganikhodzhaev, N.N., Zanin, D.V.: On a necessary condition for the ergodicity of quadratic operators defined on a two-dimensional simplex....
  • 6. Ganikhodzhaev, R.N.: A family of quadratic stochastic operators that act in S2. Dokl. Akad. Nauk UzSSR 1, 3–5 (1989)
  • 7. Ganikhodzhaev, R.N.: Quadratic stochastic operators, Lyapunov functions and tournaments. Sb. Math. 76(2), 489–506 (1993)
  • 8. Ganikhodzhaev, R., Mukhamedov, F., Rozikov, U.: Quadratic stochastic operators and processes: results and open problems. Infin. Dimens....
  • 9. Jamilov, U.U.: Quadratic stochastic operators corresponding to graphs. Lobachevskii J. Math. 34(2), 148–151 (2013)
  • 10. Jamilov, U.U.: On a family of strictly non-Volterra quadratic stochastic operators. J. Phys. Conf. Ser. 697, 012013 (2016)
  • 11. Jamilov, U.U.: On symmetric strictly non-Volterra quadratic stochastic operators. Discontin. Nonlinear Complex 5(3), 263–283 (2016)
  • 12. Jamilov, U.U., Ladra, M.: Non-ergodicity of uniform quadratic stochastic operators. Qual. Theory Dyn. Syst. 15(1), 257–271 (2016)
  • 13. Jamilov, U.U., Ladra, M., Mukhitdinov, R.T.: On the equiprobable strictly non-Volterra quadratic stochastic operators. Qual. Theory Dyn....
  • 14. Jamilov, U.U., Scheutzow, M., Wilke-Berenguer, M.: On the random dynamics of Volterra quadratic operators. Ergod. Theory Dyn. Syst. 37(1),...
  • 15. Kesten, H.: Quadratic transformations: a model for population growth: I. Adv. Appl. Probab. 2, 1–82 (1970)
  • 16. Khamraev, AYu.: On the dynamics of a quasistrictly non-Volterra quadratic stochastic operator. Ukr. Math. J. 71(8), 1116–1122 (2019)
  • 17. Lyubich, Y.I.: Mathematical structures in population genetics. In: Biomathematics, vol. 22. Springer, Berlin (1992)
  • 18. Rozikov, U.A., Zhamilov, U.: F-quadratic stochastic operators. Math. Notes 83(3–4), 554–559 (2008)
  • 19. Rozikov, U.A., Zhamilov, U.U.: Volterra quadratic stochastic operators of a two-sex population. Ukr. Math. J. 63(7), 1136–1153 (2011)
  • 20. Saburov, M.: A class of nonergodic Lotka-Volterra operators. Math. Notes 97(5), 759–763 (2015)
  • 21. Ulam, S. M.: A collection of mathematical problems. In: Interscience Tracts in Pure and Applied Mathematics, no. 8. Interscience Publishers,...
  • 22. Vallander, S.S.: On the limit behavior of sequences of iterations of some quadratic transformations. Dokl. Akad. Nauk SSSR 202(3), 515–517...
  • 23. Vallander, S. S.: On “ergodic properties of a family of quadratic stochastic operators” in rings and modules. In: Limit Theorems of Probability...
  • 24. Vallander, S.S.: Change in the orbit orientation for a certain family of stochastic quadratic mappings. Vest. St. Petersb. Univ. Math....
  • 25. Zakharevich, M.I.: On the behaviour of trajectories and the ergodic hypothesis for quadratic mappings of a simplex. Russ. Math. Surv....
  • 26. Zhamilov, U.U., Rozikov, U.A.: On the dynamics of strictly non-Volterra quadratic stochastic operators on a two-dimensional simplex. Sb....