Integrability and non-integrability of periodic non-autonomous Lyness recurrences

TL;DR

This paper investigates the dynamics of non-autonomous Lyness recurrences with periodic coefficients, identifying conditions for integrability and chaos depending on the period of the coefficient sequence.

Contribution

It classifies the integrability of Lyness recurrences based on the period of the coefficient sequence, revealing simple behavior for certain periods and complex chaos for others.

Findings

Sequences are integrable for periods 1, 2, 3, 6.

Sequences exhibit chaotic behavior for other periods.

Special features occur when the period is a multiple of 5.

Abstract

This paper studies non-autonomous Lyness type recurrences of the form $x_{n+2}=(a_n+x_{n+1})/x_{n}$, where $\{a_n\}$ is a $k$-periodic sequence of positive numbers with primitive period $k$. We show that for the cases $k\in\{1,2,3,6\}$ the behavior of the sequence $\{x_n\}$ is simple (integrable) while for the remaining cases satisfying this behavior can be much more complicated (chaotic). We also show that the cases where $k$ is a multiple of 5 present some different features.