Rich dynamical behaviors from a digital reversal operation

Abstract

An operation that maps one natural number to another can be considered as a dynamical system in $\mathbb{N}^+$. Some of such systems, e.g. the mapping in the so-called 3x+1 problem proposed by Collatz, is conjectured to have a single global attractor, whereas other systems, e.g. linear congruence, could be ergodic. Here we demonstrate that an operation that is based on digital reversal, has a spectrum of dynamical behaviors, including 2-cycle, 12-cycle, periodic attractors with other cycle lengths, and diverging limiting dynamics that escape to infinity. This dynamical system has infinite number of cyclic attractors, and may have unlimited number of cycle lengths. It also has potentially infinite number of diverging trajectories with a recurrent pattern repeating every 8 steps. Although the transient time before settling on a limiting dynamics is relatively short, we speculate that transient times may not have an upper bound.