Structured Time-Delay Models for Dynamical Systems with Connections to Frenet-Serret Frame

Abstract

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.