Hankel Alternative Model: Decomposing Chaotic Systems into Linear Dynamics with Stochastic Forcing

This paper introduces the 'Hankel alternative model', a method for decomposing chaotic systems into a linear dynamical system with a stochastic forcing term. A time series x(t) is stacked into a Hankel matrix H. The SVD of H yields a hierarchy of eigen time series that produce a delay-embedded attractor. A best-fit linear regression model is obtained on the delay coordinates v; the linear fit for the first r-1 variables is excellent, but the last coordinate vr is not well-modeled as linear. Instead, vr(t) is a stochastic input that forces the first r-1 variables. The rare events in the forcing correspond to lobe switching in the chaotic dynamics. This architecture is called the 'Hankel alternative model'.