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Dynamical systems of an infinite-dimensional non-linear operator

U. R. Olimov, U. A. Rozikov

Abstract

We investigate discrete-time dynamical systems generated by an infinite-dimensional non-linear 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.

Dynamical systems of an infinite-dimensional non-linear operator

Abstract

We investigate discrete-time dynamical systems generated by an infinite-dimensional non-linear operator that maps the Banach space 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.
Paper Structure (11 sections, 21 theorems, 113 equations, 1 figure)

This paper contains 11 sections, 21 theorems, 113 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and the main problem
  3. The case $L_1=L_2=L$
  4. Fixed points
  5. Invariant sets
  6. Dynamics on the invariant set $M_0$
  7. On the invariant set $M_{-}$
  8. The set of limit points of trajectory
  9. Jacobian matrix
  10. A sufficient condition of convergence to $M_0$
  11. Limit points

Key Result

Theorem 1

Let $X \subset \mathbb R$ and $f$ be continuously differentiable on $X$. Let $x^*\in X$ be a hyperbolic fixed point of $f$ then

Figures (1)

  • Figure 1: Graphs of $x=\psi(y)$ (green) and $y=\psi(x)$ (red) in case of having 7 fixed points. The sets mentioned in Proposition \ref{['psi']} can be seen between red and green curves.

Theorems & Definitions (37)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Remark 1
  • Lemma 1
  • Lemma 2
  • Remark 2
  • Lemma 3
  • Theorem 2
  • ...and 27 more