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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


On Quadratic Stochastic Operators Corresponding to Cyclic Groups

Discontinuity, Nonlinearity, and Complexity 6(2) (2017) 147--164 | DOI:10.5890/DNC.2017.06.003

U.A. Rozikov; U.U. Jamilov

Institute of Mathematics, National University of Uzbekistan, 29, Do’rmon Yo’li str., 100125, Tashkent, Uzbekistan

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We introduce a new class of quadratic stochastic operators corresponding to cyclic groups. We study the set of fixed points and prove that almost all (w.r.t. Lebesgue measure) trajectories of such operators converge to the center of the simplex. For the cyclic groups of order 2n we show that for any subgroup corresponding quadratic stochastic operator is a regular operator.


U. Rozikov is particularly supported by Kazakhstan Ministry of Education and Science, grant 0828/GF4: “Algebras, close to Lie: cohomologies, identities and deformations”.


  1. [1]  Bernstein, S. N. (1942), Solution of a mathematical problem connected with the theory of heredity, Annals of Mathematical Statistics, 13, 53-61.
  2. [2]  Lyubich, Yu.I. (1992),Mathematical structures in population genetics. Biomathematics, 22, Springer, New-York.
  3. [3]  Ganikhodzhaev, R. N. (1993), Quadratic stochastic operators, Lyapunov functions, and tournaments. Sbornik: Mathematics, 76(2), 489-506.
  4. [4]  Ganikhodzhaev, R. N. (1994), Map of fixed points and Lyapunov functions for a class of discrete dynamical systems. Mathematical Notes, 56(5), 1125-1131 .
  5. [5]  Ganikhodzhaev, R.N. and Eshmamatova, D.B. (2006), Quadratic automorphisms of a simplex and the asymptotic behavior of their trajectories, Vladikavkaz Mathematical Journal, 8(2), 12-28.
  6. [6]  Jamilov, U.U. and Rozikov, U.A. (2009), The dynamics of strictly non-Volterra quadratic stochastic operators on the two-deminsional simplex. Sbornik: Mathematics, 200(9), 1339-1351.
  7. [7]  Rozikov, U.A. and Jamilov, U.U. (2008), F-quadratic stochastic operators. Mathematical Notes, 83(4), 554-559.
  8. [8]  Rozikov, U. A. and Shamsiddinov, N. B. (2009), On non-Volterra quadratic stochastic operators generated by a product measure. Stochastic Analysis and Applications, 27(2), 353-362.
  9. [9]  Rozikov, U.A. and Zada, A. (2012), l-Volterra Quadratic Stochastic Operators: Lyapunov Functions, Trajectories. Applied Mathematics and Information Sciences, 6(2), 329-335.
  10. [10]  Ganikhodzhaev, N.N. (2000), An apllication of the theory of Gibbs distributions to mathematical genetics. Doklady Mathematics, 61(3), 321-323.
  11. [11]  Ganikhodzhaev, N.N. and Rozikov, U.A. (2006), On quadratic stochastic operators generated by Gibbs distributions. Regular and Chaotic Dynamics, 11(4), 467-473.
  12. [12]  Ganikhodzhaev,N.N., Saburov, M.Kh., and Jamilov, U.U. (2013),Mendelian and Non-MendelianQuadratic Operators. Applied Mathematics and Information Sciences, 7(5), 1721-1729.
  13. [13]  Ganikhodzhaev, R.N., Mukhamedov, F.M. and Rozikov, U.A. (2011), Quadratic stochastic operators and processes: results and open problems, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 14(2), 279-335.
  14. [14]  Kesten, H. (1970), Quadratic transformations: a model for population growth. I. Advances in Applied Probability, 2(1), 1-82.
  15. [15]  Kesten, H. (1970), Quadratic transformations: a model for population growth. II. Advances in Applied Probability, 2(2), 179-228.
  16. [16]  Hofbauer, J. and Sigmund, K. (1988), The theory of evolution and dynamical systems. Mathematical aspects of selection. London Mathematical Society Student Texts, vol. 7, Cambridge University Press, Cambridge.
  17. [17]  Ganikhodjaev,N.N.,Wahiddin, M.R.B., and Zanin, D.V. (2008), Regularity of some class of nonlinear transformations. arXiv: math.DS/07080697.
  18. [18]  Bukhshtab, A.A. (1960), Number theory, Gos. uch.-ped. izd. MP RSFSR, (in Russian).