基于级数研究一类离散动力系统的收敛性

Studying the Convergence of a Class of Discrete Dynamical Systems Based on Series

摘要: 本文讨论了离散动力系统$\{\begin{array}{l}{x}_{n+1}=x_n+\frac{x_n^q}{n^p}\\ 0\le x_1<1\end{array}$ 的收敛性。应用级数理论，给出了系统收敛的两个充分条件。当$1<p=q$ 时，该动力系统是收敛的；当$0<p<q$ 且$q>1$ 时，该动力系统是收敛的。为更好解释相关理论，以24年阿里巴巴全球数学竞赛决赛的一道试题进行实例分析。同时，通过数值模拟的方式进一步验证了理论的正确性。

Abstract: The article discusses the convergence of the system $\{\begin{array}{l}{x}_{n+1}=x_n+\frac{x_n^q}{n^p}\\ 0\le x_1<1\end{array}$ . By applying series theory, two sufficient conditions for the system convergence are provided: condition $1<p=q$ ensures convergence, while condition $0<p<q$ and $q>1$ guarantees convergence. To better illustrate these theories, a problem from the finals of the 24th Alibaba Global Mathematics Competition is analyzed as a case study. Additionally, the correctness of the theory is further validated through numerical simulations.