Hill-Floquet Regularization MVP

What was built

hill_floquet_experiment.py implements the Fourier Hill matrix for a periodic 2D linearized system (A=I), computes its truncated spectrum, independently integrates the tangent matrix over one period to obtain the monodromy multipliers, and runs a small parameter-optimization comparison with and without a Hill penalty.

The toy Jacobian is smooth and periodic but noncommuting in time, so it is more informative than a constant diagonal test. The period is T=2*pi, with M=128 samples and default N=8.

Quantitative checks

Results are in results.json.

Constant offset scaling. Since the construction adds gamma I to the Jacobian, the exact prediction is d mu_max / d gamma = 1. The observed slope was 1.000000000004 for monodromy and 1.000000000000 for the Hill matrix. This prediction passed.
Stability boundary. The predicted boundary is rho_F=1 exactly when max Re(mu)=0. The independently integrated monodromy crossed at gamma=0.002251, but the raw N=8 Hill maximum crossed at gamma=-0.005914. This is an absolute boundary error of about 0.00816, and more importantly the Hill maximum was positive at gamma=0 even though the true system was stable (mu_max=-0.002251). This prediction failed for the raw finite-section maximum.
Fourier truncation. The raw Hill maximum was 0.008794, 0.005890, 0.005914, 0.005914, 0.005914 for N=1,2,4,8,12, while the true monodromy exponent was -0.002251. Thus N=4 to N=8 changes by only 3.3e-11, but it converges to the wrong dominant root. The apparent convergence is spectral pollution, not validation of the physical Floquet exponent.

An independent residual check using Fourier coefficients of a numerically propagated Floquet mode gave relative Hill residuals below 6e-8 for both true monodromy exponents. Hence the matrix assembly can represent the physical roots, but selecting max Re from the finite spectrum selects a nonphysical root.

Secondary optimization signal

The toy task loss prefers an unstable gamma=0.2. Baseline gives mu_Hill=0.205914 and rho=3.647. The penalty optimization moves to gamma=0.062724, with mu_Hill=0.068638 and rho=1.539. This reduces the measured raw Hill growth, but does not establish true stability and is not counted as a win.

Verdict

The mechanism's exact offset-scaling prediction manifested, and the physical Floquet modes satisfy the assembled Hill equation. However, the proposed practical criterion—penalizing the largest real eigenvalue of a truncated Hill matrix—failed in this smooth 2D example: it produced a persistent positive spectral-pollution root and shifted the stability boundary. A robust implementation needs mode selection/continuation, structure-preserving discretization, or an independently validated monodromy/descriptor treatment before using this as a regularizer.