Skip to main content
SpringerLink
Log in

The Dynamics of Superposition of Non-Volterra Quadratic Stochastic Operators on \(S^2\)

  • Conference paper
  • First Online:
Infinite Dimensional Analysis, Quantum Probability and Applications (ICQPRT 2021)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 390))

Included in the following conference series:

  • 317 Accesses

Abstract

In this paper, we study the dynamical systems generated by stochastic operators which are a superposition of non-Volterra quadratic stochastic operators defined on the two-dimensional simplex. We showed that such stochastic operator has two fixed points and there are no periodic points except fixed points. We found attractive basins for each fixed points. We constructed a Lyapunov function and using it we showed that such operator has the property being regular.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Bernstein, S.: Solution of a mathematical problem connected with the theory of heredity. Ann. Math. Stat. 13(1), 53–61 (1942). https://doi.org/10.1214/aoms/1177731642

    Article  MathSciNet  MATH  Google Scholar 

  2. Devaney, R.L.: An introduction to chaotic dynamical systems. In: Studies in Nonlinearity. Westview Press, Boulder, CO (2003)

    Google Scholar 

  3. Ganikhodjaev, N.N., Ganikhodjaev, R.N., Jamilov, U.U.: Quadratic stochastic operators and zero-sum game dynamics. Ergod. Theory Dyn. Syst. 35, 1443–1473 (2015). https://doi.org/10.1017/etds.2013.109

  4. Ganikhodjaev, N.N., Jamilov, U.U., Mukhitdinov, R.T.: Nonergodic quadratic stochastic operators for a two-sex population. Ukr. Math. J. 65(8), 1282–1291 (2014). https://doi.org/10.1007/s11253-014-0858-2

  5. Ganikhodzhaev, R.N.: A family of quadratic stochastic operators that act in \(S^2\). Dokl. Akad. Nauk UzSSR 1, 3–5 (1989)

    MathSciNet  MATH  Google Scholar 

  6. Ganikhodzhaev, R.N.: Quadratic stochastic operators, Lyapunov functions and tournaments. Sb. Math. 76, 489–506 (1993). https://doi.org/10.1070/SM1993v076n02ABEH003423

    Article  MathSciNet  MATH  Google Scholar 

  7. Ganikhodzhaev, R., Mukhamedov, F., Rozikov, U.: Quadratic stochastic operators and processes: results and open problems. Infin. Dimens. Anal. Quan. Probab. Relat. Top. 14, 279–335 (2011). https://doi.org/10.1142/S0219025711004365

    Article  MathSciNet  MATH  Google Scholar 

  8. Jamilov, U.U.: On a family of strictly non-Volterra quadratic stochastic operators. J. Phys. Conf. Ser. 697, 012013 (2016). https://doi.org/10.1088/1742-6596/697/1/012013

  9. Jamilov, U.U.: On symmetric strictly non-Volterra quadratic stochastic operators. Disc. Nonlin. Comp. 5, 263–283 (2016). https://doi.org/10.5890/DNC.2016.09.006

    Article  MATH  Google Scholar 

  10. Jamilov, U.U., Ladra, M.: Non-ergodicity of uniform quadratic stochastic operators. Qual. Theory Dyn. Syst. 15, 257–271 (2016). https://doi.org/10.1007/s12346-015-0145-0

    Article  MathSciNet  MATH  Google Scholar 

  11. Jamilov, U.U., Scheutzow, M., Wilke-Berenguer, M.: On the random dynamics of Volterra quadratic operators. Ergod. Theory Dyn. Syst. 37, 228–243 (2017). https://doi.org/10.1017/etds.2015.30

  12. Khamraev, A.Y.: On the dynamics of a quasi strictly non-Volterra quadratic stochastic operator. Ukr. Math. J. 71, 1273–1281 (2020). https://doi.org/10.1007/s11253-019-01712-w

  13. Lyubich, Y.I.: Mathematical Structures in Population Genetics. Biomathematics, vol. 22. Springer- Verlag, Berlin (1992)

    Google Scholar 

  14. Mukhamedov, F., Embong, A.F., Rosli, A.: Orthogonal preserving and surjective cubic stochastic operators. Ann. Funct. Anal. 8, 490–501 (2017). https://doi.org/10.1215/20088752-2017-0013

    Article  MathSciNet  MATH  Google Scholar 

  15. Mukhamedov, F.M., Ganikhodjaev, N.N.: Quantum Quadratic Operators and Processes. Springer, Berlin (2015). https://doi.org/10.1007/978-3-319-22837-2

  16. Mukhamedov, F.M., Jamilov, U.U., Pirnapasov, A.T.: On non-ergodic uniform Lotka-Volterra operators. Math. Notes 105, 258–264 (2019). https://doi.org/10.1134/S0001434619010280

    Article  MathSciNet  MATH  Google Scholar 

  17. Mukhamedov, F., Pah, C.H., Rosli, A.: On Non-ergodic Volterra Cubic Stochastic operators. Qual. Theory Dyn. Syst. 18, 1225–1235 (2019). https://doi.org/10.1007/s12346-019-00334-8

  18. Mukhamedov, F., Saburov, M.: Stability and monotonicity of Lotka-Volterra type operators. Qual. Theory Dyn. Syst. 16, 249–267 (2017). https://doi.org/10.1007/s12346-016-0190-3

  19. Rozikov, U.A., Zhamilov, U.: \(F\)-quadratic stochastic operators. Math. Notes 83, 554–559 (2008). https://doi.org/10.1134/S0001434608030280

    Article  MathSciNet  MATH  Google Scholar 

  20. Rozikov, U.A., Zhamilov, U.U.: Volterra quadratic stochastic operators of a two-sex population. Ukr. Math. J. 63, 1136–1153 (2011). https://doi.org/10.1007/s11253-011-0568-y

  21. Zhamilov, U.U., Rozikov, U.A.: On the dynamics of strictly non-Volterra quadratic stochastic operators on the two-dimensional simplex. Russ. Acad. Sci. Sb. Math. 200, 1339–1351 (2009). https://doi.org/10.1070/SM2009v200n09ABEH004039

Download references

Author information

Authors and Affiliations

  1. V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 9, University Str., Tashkent, 100174, Uzbekistan

    Uygun Jamilov & Kamola Aralova

  2. Akfa University, 264, National Park Str., 111221, Qibray District, Tashkent Region, Uzbekistan

    Uygun Jamilov

  3. National University of Uzbekistan, 9, Universitet Str., 100174, Tashkent, Uzbekistan

    Uygun Jamilov

Authors
  1. Uygun Jamilov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kamola Aralova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Uygun Jamilov .

Editor information

Editors and Affiliations

  1. Vito Volterra Interdepartamental Centre, Universita degli Studi di Roma, Rome, Italy

    Luigi Accardi

  2. Department of Mathematical Sciences, College of Science, United Arab Emirates University, Al Ain, Abu Dhabi, United Arab Emirates

    Farrukh Mukhamedov

  3. Department of Mathematical Sciences, College of Science, United Arab Emirates University, Al Ain, Abu Dhabi, United Arab Emirates

    Ahmed Al Rawashdeh

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Jamilov, U., Aralova, K. (2022). The Dynamics of Superposition of Non-Volterra Quadratic Stochastic Operators on \(S^2\). In: Accardi, L., Mukhamedov, F., Al Rawashdeh, A. (eds) Infinite Dimensional Analysis, Quantum Probability and Applications. ICQPRT 2021. Springer Proceedings in Mathematics & Statistics, vol 390. Springer, Cham. https://doi.org/10.1007/978-3-031-06170-7_23

Download citation

Publish with us

Policies and ethics

Search

Navigation