Fig. 1.

The geometric meaning of DMD performed on d observables, ${\varphi }_1(\mathbf{x}),\dots ,{\varphi }_d(\mathbf{x})$, after initial fast transients die out at an exponential rate in the data. DMD then identifies the leading-order (linear) dynamics on a d-dimensional attracting spectral submanifold (SSM) $\mathcal{W}\left(E\right)$ tangent to the d-dimensional slow spectral subspace E. These linearized dynamics can be expressed in terms of the SSM-restricted observables ${\mathit{\phi }=\mathit{\varphi }|}_{\mathcal{W}\left(E\right)}$. Also shown is the spectral subspace F of the faster decaying modes and its associated nonlinear continuation, the fast spectral subspace $\mathcal{W}\left(F\right)$