Figure 3.1 is a copy of QT's Figure 6 (copied with permission).
Figure 3.1: Data from QT's Figure 6, used with permission. Their caption was: ``Results from attempts to shadow noisy orbits in the unsoftened Plummer model of Fig. 1 [their Figure 1, not included in this thesis] (each point represents a different orbit): (a) maximum time T for which the orbit could be shadowed versus the accuracy parameter used to generate the orbit; (b) maximum number of time-steps N for which orbit could be shadowed; (c) shadow distance versus the maximum one-step phase-space error in the noisy orbit; (d) versus . The dashed lines are least-squares fits to the data.'' See text for a possible explanation of why (b) has less scatter than (a).
In their text, they mention that part (b) of their figure was included simply because the scatter was less when they plotted shadow length in timesteps rather than absolute time. (Recall that they used every internal timestep of their integrator as a shadow step; thus the shadow steps would be shorter near close encounters.) They could not provide an adequate explanation of why the scatter was less in such a graph. What follows is a tentative explanation.
First, I assume that there are two distinct error magnification processes effective in N-body systems: one is magnification due to the global potential, and the other, a much higher magnification that acts much less frequently, is due to collisional encounters. Thus, if we were to plot successive times at which a doubling (or e-folding) occurs, the inter-doubling times would be shorter near close encounters than far from them. Second, different solutions obviously have their close encounters at different times in their evolutions. Third, define pseudo-time to progress in units such that the changing doubling times occur at constant intervals in pseudo-time. The result is that pseudo-time speeds up near close encounters, in comparison to real time. Since glitches occur more frequently near close encounters in real time, glitches will occur at a more uniform rate in pseudo-time. Finally, the dynamic timesteps used by QT's integrator will more uniformly follow pseudo-time than does real time, since it uses small timesteps near close encounters. In other words, the scatter of glitch occurrence (i.e., end of a shadow that follows the noisy orbit) is less in pseudo-time (and therefore in integration timesteps) than in real time. This may explain why QT's graph 6(b) has less scatter than their graph 6(a).