PyDMD: A Python package for robust dynamic mode decomposition

The paper’s example matches the exact setup X(x_i,t_k)=sech(x_i+3)cos(2.3 t_k)+2 sech(x_i) tanh(x_i) sin(2.8 t_k) and explicitly states (i) one needs r=4 modes because the data comprise two real oscillatory components (requiring two conjugate pairs) and (ii) the rank increases from two to four after applying time-delays, as shown in their demonstration with d=10 delays . The candidate solution proves rank(X)=2, gives conditions under which rank(X_d)=4 generically, and justifies r≥4 under purely imaginary, conjugate-pair constraints; these conclusions align with the paper and with their constrained BOP-DMD code snippet . There is a minor technical flaw in the model’s Step 2: asserting rank([H(c); H(s)]) = rank(H(c)) + rank(H(s)) for vertical stacking is not generally valid. However, the claimed rank 4 still follows (and can be repaired) via a simple outer-product decomposition using the four exponentials e^{±iαt}, e^{±iβt}.