U. Rozikov
29, Khodzhaeva st., 700143, Tashkent, Uzbekistan
Romanovskii Mathematical Institute of the National Academy of Sciences of the Uzbek Republic
Publications:
Olimov U. R., Rozikov U. A.
Dynamical Systems of an Infinite-Dimensional Nonlinear Operator on the Banach Space $l_1$
2024, Vol. 20, no. 4, pp. 685-703
Abstract
We investigate discrete-time dynamical systems generated by an infinite-dimensional nonlinear operator that maps the Banach space $l_1^{}$ to itself. It is demonstrated that this operator possesses up to seven fixed points. By leveraging the specific form of our operator, we illustrate that analyzing the operator can be simplified to a two-dimensional approach. Subsequently, we provide a detailed description of all fixed points, invariant sets for the two-dimensional operator and determine the set of limit points for its trajectories. These results are then applied to find the set of limit points for trajectories generated by the infinite-dimensional operator.
|
Diyorov A. M., Rozikov U. A.
Evolutionary Behavior in a Two-Locus System
2023, Vol. 19, no. 3, pp. 297-302
Abstract
In this short note we study a dynamical system generated by a two-parametric quadratic
operator mapping a 3-dimensional simplex to itself. This is an evolution operator of the frequencies
of gametes in a two-locus system. We find the set of all (a continuum set of) fixed points and
show that each fixed point is nonhyperbolic. We completely describe the set of all limit points of
the dynamical system. Namely, for any initial point (taken from the 3-dimensional simplex) we
find an invariant set containing the initial point and a unique fixed point of the operator, such
that the trajectory of the initial point converges to this fixed point.
|