Classification and Dynamics of Class of ξ(as)-QSO

Authors

  • Izzat Qaralleh Department of Mathematics, Faculty of Science, Tafila Technical University, Tafila 66110
  • Ahmad Termimi Ab Ghani Department of Mathematics, Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, 21030 Kuala Nerus, Terengganu
  • Hamza Abd Hamza Abd El-Qader Department of Mathematics, Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, 21030 Kuala Nerus, Terengganu

DOI:

https://doi.org/10.5614/j.math.fund.sci.2020.52.2.2

Keywords:

fixed point, limiting point, quadratic stochastic, periodic point, x (as)-QSO

Abstract

The current study provides a new class of ξ(as)-QSO defined on 2D simplex and classifies it into 18 non-conjugate (isomorphic) classes. This classification is based on their conjugacy and the remuneration of coordinates. The current study also examines the limiting points associated with the behavior of trajectories for four classes defined on 2D simplex.

References

Bernstein, S., Solution of a Mathematical Problem Connected with the Theory of Heredity, Annals of Math. Statis. 13, pp. 53-61, 1942.

Plank, M. & Losert, V., Hamiltonian Structures for The N-Dimensional Lotka-Volterra Equations, J. Math. Phys., 36, pp. 3520-3543, 1995.

Udwadia, F.E. & Raju, N., Some Global Properties of a Pair of Coupled Maps: Quasi-Symmetry, Periodicity and Synchronicity, Physics D, 111, pp. 16-26, 1998.

Hofbauer, J., Hutson, V. & Jansen, W., Coexistence for Systems Governed by Difference Equations of Lotka-Volterra Type, Jour. Math. Biology, 25, pp. 553-570, 1987.

Hofbauer, J. & Sigmund, K., The Theory of Evolution and Dynamical Systems, Mathematical Aspects of Selection, Cambridge Univ. Press, 1988.

Dohtani, A., Occurrence of Chaos in Higher-dimensional Discrete-time Systems, SIAM J. Appl. Math., 52(6), pp. 1707-1721, 1992.

Lyubich, Y.I., Mathematical Structures in Population Genetics, Springer-Verlag, 1992.

Kesten, H., Quadratic Transformations: A Model for Population Growth, I, II, Appl. Probab., 201, pp. 1-82, 1970.

Ulam, S.M., Problems in Modern Math., John Wiley & Sons, New York, USA, 1964.

Rozikov, U.A. & Zhamilov, U.U., On F-Quadratic Stochastic Operators, Math. Notes., 83, pp. 554-559, 2008.

Ganikhodzhaev, R.N., Quadratic Stochastic Operators, Lyapunov Functions and Tournaments, Russian Academy of Science, Sbornik Math., 76(2), pp. 489-506, 1993.

Ganikhodzhaev, R.N., A Chart of Fixed Points and Lyapunov Functions for A Class of Discrete Dynamical Systems, Math. Notes, 56, pp. 1125-1131, 1994.

Ganikhodzhaev, R.N. & Eshmamatova, D.B., Quadratic Automorphisms of A Simplex and the Asymptotic Behavior of Their Trajectories, Vladikavkaz. Math. Us., 8(2), pp. 12-28, 2006. (Text in Russian)

Jenks, R.D, Quadratic Differential Systems for Interactive Population Models, Journal of Differential Equations, 5, pp. 497-514, 1969.

Ganikhodzhaev, R.N. & Dzhurabaev, A.M., The Set of Equilibrium States of Quadratic Stochastic Operators of Type V_, Uzbek Math. Jour., 3, pp. 23-27, 1998. (Text in Russian)

Ganikhodzhaev, R.N. & Abdirakhmanova, R.E., Description of Quadratic Automorphisms of a Finite-Dimensional Simplex, Uzbek. Math. Jour, 1, pp. 7-16, 2002. (Text in Russian)

Rozikov, U.A. & Zada, A., On l-Volterra Quadratic Stochastic Operators, Inter. Journal Biomath., 3, pp. 143-159, 2010.

Rozikov, U.A. & Zada, A., l-Volterra Quadratic Stochastic Operators: Lyapunov Functions, Trajectories, Appl. Math. & Infor. Sci., 6, pp. 329-335, 2012.

Ganikhodzhaev N.N. & Mukhitdinov R.T., On A Class of Measures Corresponding to Quadratic Operators, Dokl. Akad. Nauk Rep. Uzb, 3, pp. 3-6, 1995. (Text in Russian)

Ganikhodzhaev, R.N., A Family of Quadratic Stochastic Operators that Act in S2, Dokl. Akad., Nauk UzSSR, 1(35), 1989. (Text in Russian)

Stein, P.R. & Ulam, S.M., Non-Linear Transformation Studies On Electronic Computers, Los Alamos Scientific Lab., N. Mex, 1962.

Rozikov, U.A. & Zhamilov, U.U., On Dynamics of Strictly Non-Volterra Quadratic Operators Defined On the Two Dimensional Simplex, Sbornik Math., 9, pp. 81-94. 2009. doi:10.1070/SM2009V200N09ABEH004039

Ganikhodzhaev, N.N. & Rozikov, U.A., On Quadratic Stochastic Operators Generated by Gibbs Distributions, Regular and Chaotic Dynamics, 11(4), pp. 467-473, 2006.

Ganikhodjaev, N.N., An Application of the Theory of Gibbs Distributions to Mathematical Genetics, Doklady Math., 61, pp. 321-323, 2000.

Rozikov, U.A. & Shamsiddinov, N.B., On Non-Volterra Quadratic Stochastic Operators Generated by A Product Measure, Stochastic Analysis and Applications, 27(2), pp. 353-362, 2009.

Mukhamedov, F. & Ganikhodjaev, N., Quantum Quadratic Operators and Processes, Springer, Berlin, 2015.

Mukhamedov, F. & Jamal, A.H.M., On 3/4^((s))-Quadratic Stochastic Operators in 2-Dimensional Simplex, Proc. the 6th IMT-GT Conf. Math., Statistics and its Applications (ICMSA2010), Kuala Lumpur, 3-4 November 2010, Universiti Tunku Abdul Rahman, Malaysia, pp. 159-172, 2010.

Mukhamedov, F., Saburov, M. & Qaralleh, I., On 3/4^((s))-quadratic Stochastic Operators on Two-Dimensional Simplex and Their Behavior, Abstract and Applied Analysis, 2013, pp. 1-13. 2013.

Alsarayreh, A., Qaralleh, I., Ahmad, M.Z., Al-Shutnawi, B. & Al-Kaseasbeh, S., Global and Local Behavior of a Class of 3/4^((s))-QSO, Journal of Non-linear Science, 10(9), pp. 4834-4845, 2017.

Mukhamedov F., Qaralleh, I. & Rozali W.N.F.A.W., On 3/4^((a))-Quadratic Stochastic Operators on 2-D Simplex, Sains Malaysiana, 43(8), pp. 1275-1281, 2014.

Qaralleh, I. Classification of a New Subclass of 3/4^((s))-QSO and Its Dynamics, Journal of Mathematics and Computer Science, 17(4), pp. 535-544, 2017.

Mukhamedov, F., Saburov, M. & Jamal, A.H.M., On Dynamics of 3/4^((s))-Quadratic Stochastic Operators, Inter. Jour. Modern Phys.: Conference Series 9, pp. 299-307, 2012.

Alsarayreh, A., Qaralleh, I., & Ahmad, M.Z., On 3/4^((as))-quadratic Stochastic Operators in Two-Dimensional Simplex and Their Behavior, JP Journal of Algebra, Number Theory and Applications, 39(5), pp. 737-770, 2017.

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Published

2020-09-06

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