The dynamical behavior of a rational difference equation through KAM theory

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.