Fig. 1. Embedding of the 100 points sampled from a circle. (A) The result obtained with scheme S after 100 iterations; (B) the result obtained with scheme NS after 1000 iterations. Both schemes used the same α=0.001. Clearly, scheme S converges to the correct configuration quickly, while NS does not. The circle shown in (A) is a stationary pattern. The pattern (B) is actually not the final one; eventually NS results tend to a more symmetric three-leaved clover-like pattern that actually rigidly rotates slowly. That is, the ‘ω-limit set’ is a sort of limit cycle.

Fig. 2. For scheme S the circle, i.e., the correct geometrical result, in Fig. 1 is a stable fixed point, but for scheme NS this correct figure is not a stable fixed point. Shown in this figure are residual errors by various schemes as a function of the number of iterations. The initial configurations were prepared from the correct result by displacing the point positions with uniform random noise of various amplitudes. The curves NS1, NS2, NS3 and NS4 show the residual errors for the initial condition with the displacement noise amplitudes 0.25, 0.05, 0.025, and 0.01, respectively. The error level eventually attained corresponds to the pattern similar to (B) in Fig. 1. In comparison, a result due to scheme S for a displacement noise amplitude 0.05 is also given as curve S. Notice that the dynamical system under study is not a continuous dynamical system, but with finite increments, so even for scheme S there is a residual error due to this quantization noise. That is why the curve S does not go to zero even asymptotically.