Dual Nature of Orbits of the Divide-or-Choose 2 Rule: A Quadratic Collatz-type Recursion

TL;DR

This paper fully characterizes the orbits of a quadratic Collatz-type recursion, identifying initial values leading to cycles or divergence, thereby advancing understanding of such nonlinear dynamical systems.

Contribution

It provides a complete classification of orbits for the divide-or-choose-2 rule, detailing conditions for periodicity and divergence.

Findings

Initial values leading to periodic orbits are explicitly characterized.

All other initial values result in orbits diverging to infinity.

The period of cycles depends on the initial value.

Abstract

We obtain a complete characterization of all orbits of a quadratic Collatz-type recursion called the divide-or-choose-2 rule. Each orbit either ends in a cycle whose period depends on the initial value or it goes to infinity. We specify which initial values generate periodic orbits, and also show that all other initial values generate orbits that go to infinity.