Quasi-Periodic Orbits of a Difference Equation with Varied Coefficients

In this paper, we study the stability of the equilibrium point and investigate the periodicity properties of a class of difference equations [Formula: see text], [Formula: see text] whose parameters and initial values are positive. In particular, [Formula: see text] is a two-periodic coefficient with respect to n. Moreover, we depict the phase transition of orbits and analyze the deviation of the quasi-periodic orbits from some periodic orbits related to the arithmetic property of [Formula: see text], including Diophantine and Liouvillean numbers. We associate the response solution problem of differential equations with the fixed point problem of the area-preserving mapping related to multiperiodic difference equations, and pose some future work. The techniques are based on Moser’s twist mapping theorem, reversibility and some invariant structures embedded in the system.