Rich dynamical behaviors from a digital reversal operation

TL;DR

This paper investigates a digital reversal operation as a dynamical system, revealing complex behaviors such as cycles and diverging trajectories, and compares it to well-known mathematical problems like the Collatz map.

Contribution

It introduces a new digital reversal map, analyzes its dynamics, and identifies novel behaviors including cycles and divergence, connecting it to existing mathematical problems.

Findings

Identification of limiting cycles with unusual periodicity

Discovery of length-8 diverging trajectories

Comparison to Collatz and reverse-add-then-sort (RATS) maps

Abstract

Repeatedly adding or subtracting the digital reversal to or from an integer, depending on which one is larger, can be treated as a dynamical system. On one hand, a three-digit version of this map running only two steps is the 1089 mathematical trick problem; on the other hand, this mapping can be compared to John Conway's reverse-add-then-sort (RATS) iteration, as well as the 3x+1 problem, also known as Collatz's map. We numerically run this map and find interesting dynamics, including limiting cycles with unusual periodicity and length-8 diverging trajectories.