The dynamical behavior of a rational difference equation through KAM theory

Ruiqi Yan; Qianhong Zhang; Ruiqi Yan; Qianhong Zhang

doi:10.3934/math.2025815

AIMS Mathematics

2025, Volume 10, Issue 8: 18252-18267. doi: 10.3934/math.2025815

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The dynamical behavior of a rational difference equation through KAM theory

Ruiqi Yan ,
Qianhong Zhang ^,

School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China

Received: 16 June 2025 Revised: 30 July 2025 Accepted: 05 August 2025 Published: 13 August 2025
MSC : 39A30

In this study, we employ the Kolmogorov-Arnold-Moser (KAM) theory to investigate the stability of solutions for the following difference equation:
$ \begin{equation*} \begin{aligned} \zeta_{n+1}& = \frac{1}{\zeta_{n-1}-\beta \zeta_n\zeta_{n-1}}, n = 0, 1, ... \end{aligned} \end{equation*} $
In this context, the parameter $ \beta $ and the initial conditions $ \zeta_{-1}, \zeta_{0} $ are positive. We construct a novel logarithmic transformation that satisfies the area-preserving mapping to demonstrate the existence of a positive elliptic equilibrium and examine the stability. Then, the Birkhoff normal form locally simplifies the nonlinear system into a higher-order standard form near the equilibrium or periodic orbits. Consequently, we apply the KAM theory to prove that there exist numerous invariant closed curves in the smooth invariant region of any non-degenerate elliptic fixed point. Additionally, we conduct multiple numerical simulations to support our research findings using the Matlab software.
- KAM theory,
- Birkhoff normal form,
- difference equation,
- area preserving map
Citation: Ruiqi Yan, Qianhong Zhang. The dynamical behavior of a rational difference equation through KAM theory[J]. AIMS Mathematics, 2025, 10(8): 18252-18267. doi: 10.3934/math.2025815

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Abstract

In this study, we employ the Kolmogorov-Arnold-Moser (KAM) theory to investigate the stability of solutions for the following difference equation:

$ \begin{equation*} \begin{aligned} \zeta_{n+1}& = \frac{1}{\zeta_{n-1}-\beta \zeta_n\zeta_{n-1}}, n = 0, 1, ... \end{aligned} \end{equation*} $

In this context, the parameter $ \beta $ and the initial conditions $ \zeta_{-1}, \zeta_{0} $ are positive. We construct a novel logarithmic transformation that satisfies the area-preserving mapping to demonstrate the existence of a positive elliptic equilibrium and examine the stability. Then, the Birkhoff normal form locally simplifies the nonlinear system into a higher-order standard form near the equilibrium or periodic orbits. Consequently, we apply the KAM theory to prove that there exist numerous invariant closed curves in the smooth invariant region of any non-degenerate elliptic fixed point. Additionally, we conduct multiple numerical simulations to support our research findings using the Matlab software.

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