On periodic solutions of 2-periodic Lyness difference equations

TL;DR

This paper investigates the existence and characteristics of periodic solutions in non-autonomous 2-periodic Lyness difference equations, revealing that most even periods occur and odd periods appear when parameters differ.

Contribution

It proves that for parameters other than (1,1), infinitely many initial conditions produce periodic sequences with almost all even periods and certain odd periods.

Findings

For (a,b) ≠ (1,1), infinitely many initial conditions lead to periodic sequences.

Almost all even periods are realized in these sequences.

When a ≠ b, all odd periods except 1 can occur.

Abstract

We study the existence of periodic solutions of the non--autonomous periodic Lyness' recurrence u_{n+2}=(a_n+u_{n+1})/u_n, where {a_n} is a cycle with positive values a,b and with positive initial conditions. It is known that for a=b=1 all the sequences generated by this recurrence are 5-periodic. We prove that for each pair (a,b) different from (1,1) there are infinitely many initial conditions giving rise to periodic sequences, and that the family of recurrences have almost all the even periods. If a is not equal to b, then any odd period, except 1, appears.