The eigenvalues of the Koopman matrix K of synthetic Duffing oscillator data. The x-axis values are the real parts of the eigenvalues, and the y-axis values are the imaginary parts of the eigenvalues. The dashed curve marks the unit circle. We highlight two example eigenvalues: the red circle marks one of the eigenvalues with a small phase angle θ (a slow-varying mode), and the green circle marks one of the eigenvalues with a larger phase angle θ (a fast-oscillating mode). Their corresponding eigenfunctions are labeled φ_slow and φ_fast.