On two and three periodic Lyness difference equations

TL;DR

This paper investigates 2- and 3-periodic non-autonomous Lyness difference equations, revealing unique oscillatory behaviors in their associated rotation number functions not seen in autonomous cases.

Contribution

It characterizes the dynamics of non-autonomous Lyness equations with periodic coefficients and uncovers novel oscillation phenomena in their rotation number functions.

Findings

Existence of a single oscillation in the rotation number functions.

Distinct behavior from autonomous Lyness difference equations.

Analysis of sequences with 2-periodic and 3-periodic coefficients.

Abstract

We describe the sequences {x_n}_n given by the non-autonomous second order Lyness difference equations x_{n+2}=(a_n+x_{n+1})/x_n, where {a_n}_n is either a 2-periodic or a 3-periodic sequence of positive values and the initial conditions x_1,x_2 are as well positive. We also show an interesting phenomenon of the discrete dynamical systems associated to some of these difference equations: the existence of one oscillation of their associated rotation number functions. This behavior does not appear for the autonomous Lyness difference equations.