Pilot-Wave Dynamics: Using Dynamic Mode Decomposition to characterize Bifurcations, Routes to Chaos and Emergent Statistics

The paper documents, via optimized DMD, a progression with increasing forcing Γ from a steady wave (rank-1, eigenvalue at the origin) to periodic (rank-3, complex-conjugate pair indicative of Hopf), to secondary period-doubling with approximately harmonic pairs (rank-5), and finally to complex, chaotic dynamics with incommensurate frequencies (rank-9) . The authors explicitly present a period-doubling route to chaos in the simulations and summarize a series of Hopf onsets culminating in a period-doubling cascade to spatio-temporal chaos . The model’s Part A interprets the same DMD spectra through the lens of standard local bifurcation theory (flip/Neimark–Sacker) and makes the stroboscopic Poincaré-map connection; this is consistent with the paper’s narrative but somewhat more formal and slightly more specific (e.g., period-2 and period-4 claims require assuming exact strobing at TF and identifying discrete-time angles), which the paper does not quantify. For Part B, the paper sets up the inverse problem η = Gμ with constraints μ ≥ 0, ||Δx μ||1 = 1, notes severe ill-conditioning and a large approximate nullspace, and solves it numerically (NN-FCGLS), arguing the constraints act as regularizers . The model provides a rigorous existence result on the simplex and basic stability bounds under a restricted singular value condition; these augment, rather than contradict, the paper’s qualitative discussion. Hence, both are correct, with the model offering a more proof-oriented framing while the paper focuses on empirical DMD evidence and numerical inversion.