Discontinuity, Nonlinearity, and Complexity
Dynamical Systems Generated by a Gonosomal Evolution Operator
Discontinuity, Nonlinearity, and Complexity 5(2) (2016) 173185  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 discretetime 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 AixMarseille 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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