N Bugs on a Circle

TL;DR

This paper generalizes the classic pursuit problem to bugs on a circle, revealing diverse steady states and analyzing their stability, with exact and estimated probabilities for different configurations.

Contribution

It introduces a new circular pursuit model, analyzes its steady states, and provides probabilistic insights into long-term behaviors for various bug counts.

Findings

Three types of steady states identified: coalescence, antipodal clustering, and stable cycles.

Exact probabilities derived for N ≤ 4 bugs; Monte Carlo estimates for larger N.

Coalescence probability decreases approximately as inverse square root of N.

Abstract

We describe and analyze a generalization of the classic ``Four Bugs on a Square'' cyclic pursuit problem. Instead of allowing the bugs to spiral towards one another, we constrain $N$ bugs to the perimeter of the unit circle. Depending on their configuration, each bug moves either clockwise or counterclockwise with a constant angular speed, or remains stationary. Unlike the original problem where bugs always coalesce, this generalization produces three possible steady states: all bugs coalescing to a single point, clusters of bugs located at two antipodal points, or bugs entering a stable infinite chase cycle where they never meet. We analyze the stability of these steady states and calculate the probability that randomly initialized bugs reach each state. For $N \leq 4$, we derive exact analytical expressions for these probabilities. For larger values, we employ Monte Carlo simulations to estimate the probability of coalescing, finding it approximately follows an inverse square root relationship with the number of bugs. This generalization reveals rich dynamical behaviors that are absent in the classic problem. Our analysis provides insight into how restricting the bugs to the circle's perimeter fundamentally alters the long-term behavior of pursuing agents compared to unrestricted pursuit problems.