(a) a = 1 (b) a = 1.002
Figure 6: Phase-space visualizations of standard dynamics on circular (a) and slightly perturbed circular table (b) show the
emergence of qualitatively different trajectories when changing parameter a.
was not true for symplectic dynamics, where phase-
space visualization for the perturbed table remained
qualitatively equivalent to Figure 6(a). When increas-
ing the ellipticity of the table (e.g., further increas-
ing parameter a), the phase-space visualizations for
standard dynamics exhibit more and more orbit struc-
tures, see Figure 3(a), while phase-space visualization
for symplectic dynamics still consists of horizontal
curves but with waves of increasing amplitudes, see
Figure 3(b).
As a second investigation, we looked into the im-
pact of the choice of parameters r and w by comparing
a standard elliptic table to a perturbed one. For stan-
dard dynamics, the phase-space visualizations seem
almost identical when slightly increasing r from 2 to
2.001, see Figure 4(a,b), but the comparative visu-
alization exhibits some structural changes, see Fig-
ure 4(c). In this investigation, the same holds true for
symplectic dynamics: The comparative visualization
in Figure 4(d) highlights emerging cyclic structures
for perturbed tables, which have not been observed
for the standard elliptic table (cf. Figure 3(b)).
Based on these initial investigation, we generated
ensembles with many different parameter conﬁgura-
tions. For standard dynamics, we generated initial
collisions by sampling the phase space equidistantly
with 23 position × 31 direction samples. For each
initial collision, we computed a trajectory with 200
collision points. These 23 × 31 trajectories are com-
puted for each ensemble member, where the ensem-
ble is formed by choosing parameters from ranges
a ∈ [4;4.004], b ∈ [5; 5.001], r = 2, and w ∈ [2; 2.009]
with step size 0.001. The MDS plot exhibits an arc
shape, see Figure 7. Color coding the plot to inves-
tigate the impact of the parameter, we can make the
following observations: The color transitions for pa-
rameters a, b, or their ratio a/b conﬁrm the observa-
tion from above that these parameters locally affect
the outcome, see Figure 7(b,c). The color transitions
in Figure 7(b,c) form clusters, i.e., we see groups of
points where colors go from dark to bright and then
this pattern is repeated for the next group. Investigat-
ing the inﬂuence of parameter w in Figure 7(a) pro-
vides a different picture: There is a trend of having
brighter colors to the left of the arc and darker colors
to the right, but the transition is not monotonic. How-
ever, when investigating the color transition for ratio
r/w in Figure 7(d), we observe that there is a mono-
tonic color transition from bright to dark that clearly
follows the arc. Hence, we can formulate the suppo-
sition that ratio r/w is the dominant factor for signif-
icant changes, as runs occur the more similar in the
MDS plot the more they have a similar ratio. This is
in accordance with earlier studies in the ﬁeld.
Finally, we generate a simulation ensemble for
symplectic dynamics using the same initial collisions
and trajectory lengths as above, while choosing pa-
rameters from ranges a ∈ [4; 4.006], b ∈ [6; 6.006],
r ∈ [2; 2.006], and w ∈ [2; 2.006] with step size 0.001.
Here, such studies had not been performed yet, but
is easily supported by our tool. Figure 5 shows that
parameters r and w and their ratio r/w are, again, re-
sponsible for generating main trends and clusters in
the MDS plots, while parameters a, b and their ratio
a/b are only responsible for small variations within
the clusters. Figure 5(a) clearly shows the color tran-
sitions only within each of the nine clusters for the
ratio a/b. For parameter r, there is a transition that
creates three stripes of brightness from the upper left
to the lower right. Similarly, for parameter w, there
is a transition that creates three stripes of brightness
from the lower left to the upper right. When combin-
ing the two parameters by looking at their ratio r/w,
there is a transition from left to right with ﬁve stripes.
Hence, we can conclude that for both standard and
symplectic dynamics changes of ratio r/w dominate