Figure 8.

Structured HAVOK (sHAVOK) yields more structured models from short trajectories than HAVOK. For each system, we simulated a trajectory extracting a single coordinate in time (grey). We then apply HAVOK and sHAVOK to data $x(t)$ from a short subset of this trajectory, shown in black. The middle columns show the resulting dynamics matrices $\mathit{A}$ from the models. The top three rows correspond to different subsets of the Lorenz system, while the fourth and fifth rows correspond to trajectories from the Rössler system and a double pendulum, respectively. Compared with HAVOK, the resulting models for sHAVOK consistently show stronger structure in that they are antisymmetric with non-zero elements only along the sub- and super-diagonals. The corresponding eigenvalue spectra of $\mathit{A}$ for HAVOK and sHAVOK are plotted in teal and maroon, respectively, in addition to eigenvalues from HAVOK for the full (grey) trajectory. In all cases, the sHAVOK eigenvalues are much closer in value to those from the long trajectory limit than HAVOK. (Online version in colour.)