A multichannel generalization of the HAVOK method for the analysis of nonlinear dynamical systems.

Abstract

By extending Takens' embedding theorem [Dynamical Systems and Turbulence, Warwick 1980, edited by D. Rand and L.-S. Young (Springer, Berlin, 1981), pp. 366-381], Deyle and Sugihara [PLoS One 6, 1-8 (2011)] provided a theoretical justification for using parallel measurement time series to reconstruct a system's attractor. Building on Takens' framework, Brunton et al. [Nat. Commun. 8, 19 (2017)] introduced the Hankel alternative view of Koopman (HAVOK) algorithm, a data-driven approach capable of linearizing chaotic systems through delay embeddings. In this work, a modified version of the original algorithm (mHAVOK) is presented, a practical realization of Deyle and Sugihara's generalized embedding theory. mHAVOK extends the original algorithm from one to multiple input time series and introduces a systematic approach to separating linear and nonlinear terms. An R2-informed quality score is introduced and shown to be a reliable guide for the selection of the reduced rank. The algorithm is tested on the familiar Lorenz system, as well as the more sophisticated Sprott system, which features different behaviors depending on the initial conditions. The quality of the reconstructions is assessed with the Chamfer distance, validating how mHAVOK allows for a more accurate reconstruction of the system dynamics. The new methodology generalizes HAVOK by allowing the analysis of multivariate time series, fundamental in real-life data-driven applications.