Periodic points of a Landen transformation

TL;DR

This paper proves the existence of 3-periodic orbits in a Landen transformation dynamical system, challenging previous conjectures, and introduces a systematic method to analytically find isolated periodic points in algebraic maps.

Contribution

It presents a new systematic methodology for analytically determining isolated periodic points in algebraic dynamical systems, applied to Landen transformations.

Findings

Existence of 3-periodic orbits in the Landen transformation system

Disproof of a previous conjecture on the system's dynamics

Development of a general approach for locating periodic points

Abstract

We prove the existence of 3-periodic orbits in a dynamical system associated to a Landen transformation previously studied by Boros, Chamberland and Moll, disproving a conjecture on the dynamics of this planar map introduced by the latter author. To this end we present a systematic methodology to determine and locate analytically isolated periodic points of algebraic maps. This approach can be useful to study other discrete dynamical systems with algebraic nature. Complementary results on the dynamics of the map associated with the Landen transformation are also presented.