标题： N 个虫子在圆圈上 显示英文标题

标题： N Bugs on a Circle

摘要： 我们描述并分析了经典“四只虫子在正方形上”的循环追逐问题的一个推广。 与允许虫子相互螺旋靠近不同，我们将$N$只虫子限制在单位圆的周长上。 根据它们的配置，每只虫子以恒定的角速度顺时针或逆时针移动，或者保持静止。 与原始问题中虫子总是聚合成一点不同，这个推广产生了三种可能的稳态：所有虫子聚合成一个点，虫子分布在两个相对点上形成集群，或者虫子进入一个稳定的无限追逐循环，它们永远不会相遇。 我们分析了这些稳态的稳定性，并计算了随机初始化的虫子达到每种状态的概率。 对于$N \leq 4$，我们推导出这些概率的精确解析表达式。 对于较大的值，我们采用蒙特卡洛模拟来估计聚合成一点的概率，发现它大约与虫子数量的平方根成反比。 这个推广揭示了经典问题中不存在的丰富动力学行为。 我们的分析提供了关于将虫子限制在圆周上如何从根本上改变追逐代理的长期行为的见解，相比于无约束的追逐问题。 显示英文摘要

摘要： 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.