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Discontinuity, Nonlinearity, and Complexity 5(2) (2016) 173--185 | DOI:10.5890/DNC.2016.06.007

Utkir A. Rozikov; Richard Varro

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

Institut de Mathématiques et de Modélisation de Montpellier, Université de Montpellier, 35095 Montpellier Cedex 5, France.

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Abstract

In this paper we consider discrete-time dynamical systems generated by gonosomal evolution operators of sex linked inheritance. Mainly we study dynamical systems of a hemophilia, which biologically is a group of hereditary genetic disorders that impair the body’s ability to control blood clotting or coagulation, which is used to stop bleeding when a blood vessel is broken. We give an algebraic model of the biological system corresponding to the hemophilia. The evolution of such system is studied by a nonlinear (quadratic) gonosomal operator. In a general setting, this operator is considered as a mapping from Rn, n ≥ 2 to itself. In particular, for a gonosomal operator at n = 4 we explicitly give all (two) fixed points. Then limit points of the trajectories of the corresponding dynamical system are studied. Moreover we consider a normalized version of the gonosomal operator. In the case n = 4, for the normalized gonosomal operator we show uniqueness of fixed point and study limit points of the dynamical system.

Acknowledgments

U.Rozikov thanks Aix-Marseille University Institute for Advanced Study IM´eRA (Marseille, France) for support by a residency scheme. His work also partially supported by the Grant No.0251/GF3 of Education and Science Ministry of Republic of Kazakhstan.

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