  Dynamic Mode Decomposition (DMD) is a powerful tool for extracting spatial
and temporal patterns from multi-dimensional time series, and it has been used
successfully in a wide range of fields, including fluid mechanics, robotics,
and neuroscience. Two of the main challenges remaining in DMD research are
noise sensitivity and issues related to Krylov space closure when modeling
nonlinear systems. Here, we investigate the combination of noise and
nonlinearity in a controlled setting, by studying a class of systems with
linear latent dynamics which are observed via multinomial observables. Our
numerical models include system and measurement noise. We explore the
influences of dataset metrics, the spectrum of the latent dynamics, the
normality of the system matrix, and the geometry of the dynamics. Our results
show that even for these very mildly nonlinear conditions, DMD methods often
fail to recover the spectrum and can have poor predictive ability. Our work is
motivated by our experience modeling multilegged robot data, where we have
encountered great difficulty in reconstructing time series for oscillatory
systems with slow transients, which decay only slightly faster than a period.
