Non-autonomous 2-periodic Gumovski-Mira difference equations

TL;DR

This paper investigates two types of non-autonomous 2-periodic Gumovski-Mira difference equations, revealing contrasting behaviors: one integrable and simple, the other chaotic, with a comprehensive analysis of the integrable case.

Contribution

It provides the first detailed comparison between autonomous and non-autonomous 2-periodic Gumovski-Mira equations, highlighting their differing dynamics and characterizing the integrable case.

Findings

One case exhibits simple, integrable behavior.

The other case displays chaotic dynamics.

A global analysis of the integrable case is presented.

Abstract

We consider two types of non-autonomous 2-periodic Gumovski-Mira difference equations. We show that while the corresponding autonomous recurrences are conjugated, the behavior of the sequences generated by the 2-periodic ones differ dramatically: in one case the behavior of the sequences is simple (integrable) and in the other case it is much more complicated (chaotic). We also present a global study of the integrable case that includes which periods appear for the recurrence.