Perimeter expansion in the n‐bug system and its relationship to stability

We consider a system of n bugs located at (x1,y1), ..., (xn,yn), where bug i runs away from bug i+1 with common speed v along the instantaneous line of sight. To close the cycle of flight, bug n runs away from bug 1. The computer simulation of this system indicates that random initial configurations evolve into stable regular center‐symmetric patterns—all of which have a vertex angle of less than π/2. By utilizing the Lagrange multiplier method, we show that for these stable configurations the perimeter expansion rate ṗ is a local maximum. The most stable configuration has the smallest possible vertex angle and is associated with an absolute maximum for ṗ. The regular center‐symmetric patterns with vertex angles greater than π/2 also have a stationary perimeter expansion rate. These are local minima rather than maxima, however, and belong to configurations which are unstable.