Birkhoff normal forms, KAM theory and time reversal symmetry for certain rational map (Q272099)

Summary: By using the KAM (Kolmogorov-Arnold-Moser) theory and time reversal symmetries, we investigate the stability of the equilibrium solutions of the system: \[ x_{n+1}=\frac{1}{y_n},\,y_{n+1}=\frac{\beta x_n}{1+y_n},\,n=0,1,2,\dots, \] where the parameter \(\beta>0\), and initial conditions \(x_0\) and \(y_0\) are positive numbers. We obtain the Birkhoff normal form for this system and prove the existence of periodic points with arbitrarily large periods in every neighborhood of the unique positive equilibrium. We use invariants to find a Lyapunov function and Morse's lemma to prove closedness of invariants. We also use the time reversal symmetry method to effectively find some feasible periods and the corresponding periodic orbits.