N bugs on a circle

Area of Science:

• Mathematical Physics
• Dynamical Systems
• Computational Science

Background:

• The classic "Four Bugs on a Square" problem involves cyclic pursuit where agents spiral towards each other.
• Generalizations explore variations in agent movement and environmental constraints.
• Understanding agent behavior in constrained environments is crucial for robotics and swarm intelligence.

Purpose of the Study:

• To analyze a generalization of the cyclic pursuit problem with N bugs constrained to the perimeter of a unit circle.
• To identify and characterize the possible steady states of this system.
• To calculate the probabilities of reaching each steady state for random initial configurations.

Main Methods:

• Analytical derivation of steady-state probabilities for N<=4.
• Monte Carlo simulations for N>4 to estimate coalescence probabilities.
• Stability analysis of identified steady states.

Main Results:

• Three steady states were identified: single-point coalescence, two-antipodal-point clusters, and stable infinite chase cycles.
• For N<=4, exact analytical probabilities for each state were derived.
• For larger N, coalescence probability approximates an inverse square-root relationship with N.

Conclusions:

• Constraining pursuit agents to a circle perimeter introduces complex dynamics absent in unrestricted problems.
• The system exhibits rich behaviors including stable non-coalescing states.
• This model offers insights into the long-term behavior of pursuing agents in confined spaces.