Fig. 1 | Nature Communications

Fig. 1

From: Chaos as an intermittently forced linear system

Fig. 1

Decomposition of chaos into a linear dynamical system with forcing. 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 v r is not well-modeled as linear. Instead, v r (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 view of Koopman (HAVOK) analysis