  Time-delay embeddings and dimensionality reduction are powerful techniques
for discovering effective coordinate systems to represent the dynamics of
physical systems. Recently, it has been shown that models identified by dynamic
mode decomposition (DMD) on time-delay coordinates provide linear
representations of strongly nonlinear systems, in the so-called Hankel
alternative view of Koopman (HAVOK) approach. Curiously, the resulting linear
model has a matrix representation that is approximately antisymmetric and
tridiagonal with a zero diagonal; for chaotic systems, there is an additional
forcing term in the last component. In this paper, we establish a new
theoretical connection between HAVOK and the Frenet-Serret frame from
differential geometry, and also develop an improved algorithm to identify more
stable and accurate models from less data. In particular, we show that the sub-
and super-diagonal entries of the linear model correspond to the intrinsic
curvatures in Frenet-Serret frame. Based on this connection, we modify the
algorithm to promote this antisymmetric structure, even in the noisy, low-data
limit. We demonstrate this improved modeling procedure on data from several
nonlinear synthetic and real-world examples.
