Square and bow-tie configurations in the cyclic evasion problem

Cyclic evasion of four agents on the plane is considered. There are two stationary shapes of configurations: square and degenerate bow-tie. The bow-tie is asymptotically attracting while the square is of focus-center type. Normal form analysis shows that square is nonlinearly unstable. The stable manifold consists of parallelograms that all converge to the square configuration. Based on these observations and numerical simulations, it is conjectured that any non-parallelogram non-degenerate configuration converges to the bow-tie.