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ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
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

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

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

Email: dumitru.baleanu@gmail.com

On Symmetric Strictly non-Volterra Quadratic Stochastic Operators

Discontinuity, Nonlinearity, and Complexity 5(3) (2016) 263--283 | DOI:10.5890/DNC.2016.09.006

U.U. Jamilov

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

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Abstract

For a symmetric strictly non-Volterra quadratic stochastic operator on the three-dimensional simplex it is proved that this operator has a unique fixed point. A sufficient condition of attractiveness for the unique fixed point is found. For such operators we describe the set of ω− limit points. We proved that some classes of such operators have infinitely many periodic points. Also it is shown that there are trajectories which are asymptotically cyclic with period two.

Acknowledgments

The Author grateful to professors U. A. Rozikov, J. Blath and M. Scheutzow for helpful discussions. He thanks IMU Berlin Einstein Foundation Program (EFP), Berlin Mathematical School(BMS) for scholarship and for support of his visit to Technical University(TU) Berlin and TU Berlin for kind hospitality.

References

  1. [1]  Akin, E. and Losert, V. (1984), Evolutionary dynamics of zero-sum games, Journal of Mathematical Biology, 20, 231-258.
  2. [2]  Bernstein, S.N. (1942), The solution of a mathematical problem related to the theory of heredity, Annals Mathematical Statistics, 13, 53-61.
  3. [3]  Blath, J., Jamilov(Zhamilov), U.U., and Scheutzow, M. (2014), (G,μ)-quadratic stochastic operators, Journal of Difference Equations and Applications, 20, 1258-1267.
  4. [4]  Devaney, R.L. (2003), An introduction to chaotic dynamical systems, Westview Press: New York.
  5. [5]  Elaydi, S.N. (2000), Discrete Chaos, Chapman Hall/CRC: Florida.
  6. [6]  Ganikhodjaev, N.N. (1991), Doctoral Thesis (Rehabilitation Degree), Institute for Low Temperature Physics and Engineering Academy of Sciences of Ukraine: Kharkov.
  7. [7]  Ganikhodjaev, N.N., Ganikhodjaev, R.N., and Jamilov U.U. (2015), Quadratic stochastic operators and zero-sum game dynamics, Ergodic Theory and Dynamical Systems, 35, 1443-1473.
  8. [8]  Ganikhodjaev, N.N., Jamilov, U.U., and Mukhitdinov, R.T. (2013) On Non-Ergodic Transformations on S3, Journal of Physics: Conference Series, 435, 012005.
  9. [9]  Ganikhodjaev, N.N., Zhamilov, U.U., and Mukhitdinov, R.T. (2013), Non-ergodic quadratic operators of bisexual population, Ukrainian Mathematical Journal, 65, 1152-1160.
  10. [10]  Ganikhodjaev, N.N. and Zanin, D.V. (2004) On a necessary condition for the ergodicity of quadratic operators defined on the two-dimensional simplex,Russian Mathematical Surveys, 59, 571-572.
  11. [11]  Ganikhodjaev N.N. and Zanin, D.V. Ergodic Volterra quadratic maps of the simplex, arXiv: 1205.3841. (in Russian)
  12. [12]  Ganikhodzhaev, R.N. (1993), Quadratic stochastic operators, Lyapunov functions, and tournaments, Sbornik: Mathematics, 76, 489-506.
  13. [13]  Ganikhodzhaev, R.N. (1994), Map of fixed points and Lyapunov functions for a class of discrete dynamical systems, Mathematical Notes, 56, 1125-1131 .
  14. [14]  Ganikhodzhaev, R.N. and Eshmamatova, D. B. (2006), Quadratic automorphisms of a simplex and the asymptotic behavior of their trajectories, Vladikavkaz Mathematical Journal, 8, 12-28, (in Russian).
  15. [15]  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, 279-335.
  16. [16]  Jamilov, U.U. (2008), Regularity of F-quadratic stochastic operators, Uzbek Mathematical Journal,2, 35-45, (in Russian).
  17. [17]  Jamilov, U.U. andMukhitdinov, R.T. (2010), Conditional quadratic stochastic operators, Uzbek Mathematical Journal, 2, 31-38, (in Russian).
  18. [18]  Lyubich, Yu.I. (1992),Mathematical structures in population genetics, Biomathematics, 22, Springer-Verlag, Berlin.
  19. [19]  Lyubich, Yu.I. (1974), Iterations of quadratic maps, Mathematical economy and functional analysis, 109-138, Nauka, Moscow, (Russian).
  20. [20]  Nagylaki, T. (1983), Evolution of a large population under gene conversion, Proceedings of the National Academy of Sciences of the United States of America, 80, 5941-5945.
  21. [21]  Nagylaki, T. (1983), Evolution of a finite population under gene conversion, Proceedings of the National Academy of Sciences of the United States of America, 80, 6278-6281.
  22. [22]  Robinson, R.C. (2004), An Introduction to Dynamical Systems: Continuous and Discrete, Pearson Education, New Jersey.
  23. [23]  Rozikov, U.A. and Zhamilov, U.U. (2008), F-quadratic stochastic operators, Mathematical Notes, 83, 554-559.
  24. [24]  Rozikov, U.A. and Zhamilov, U.U. (2009), The dynamics of strictly non-Volterra quadratic stochastic operators on the two-dimensional simplex, Sbornik: Mathematics, 200, 1339-1351.
  25. [25]  Rozikov, U.A. and Zhamilov, U.U. (2011), Volterra quadratic stochastic of a two-sex population, Ukrainian Mathematical Journal, 63, 1136-1153.
  26. [26]  Sharkovskii, A.N. and etc. (1989), The Dynamics of One Dimensional Maps, Naukovo Dumka, Kiev.
  27. [27]  Ulam, S.M. (1960), A Collection of Mathematical Problems, Interscience Publication, New York.
  28. [28]  Volterra, V. (1931)Variations and Fluctuations of the Number of Individuals in Animal Species Living Together in Animal Ecology, (edition Chapman, R.N., McGraw-Hill, New York).
  29. [29]  Zakharevich, M.I. (1978), On a limit behavior and ergodic hypothesis for quadratic mappings of a simplex, Russian Mathematical Surveys, 33, 207-208.

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