A Computationally Bounded Test of Periodic and Aperiodic Forcing in the Forced Kuramoto–Sivashinsky Equation

Paper 14 in The Geometry of the Critical Line programme.

We test the conjecture from Paper 13 that deterministic aperiodic spatial forcing suppresses steep-gradient formation in the forced Kuramoto–Sivashinsky (KS) equation, analogous to lock-in suppression in forced Kuramoto networks (Paper 11). Using pseudospectral simulations across nine forcing families and amplitudes F ∈ [0.5, 8.0], we obtain a robust negative result: periodic sinusoidal forcing produces the lowest time-averaged gradients, while aperiodic and random forcing preserve stronger steep-gradient activity. The contrast strengthens monotonically with forcing amplitude.

Diagnostics reveal the mechanism. Periodic forcing imposes a coherent phase-lagged spatial lock, generating a rigid low-mode hierarchy via the −½(θ_x)² nonlinear coupling. By Parseval's theorem, periodic forcing concentrates spectral energy into a narrow band sufficient to overpower intrinsic KS steepening, while aperiodic forcing dilutes the same energy across a broadband spectrum. A stick-slip regime at moderate amplitudes (F ≈ 2.0) produces intermittent violent gradient snapping despite low average gradients. Spectral entropy is higher in the periodic case (0.53 vs 0.06–0.08), tracking harmonic ladder breadth rather than macroscopic disorder; this ordering is verified robust under noise-floor thresholds from 10⁻¹² to 10⁻².

All claims are bounded to the numerically stable integration window (t ≤ 100, N = 256), with Lyapunov time τ_c ≪ 100 anchoring the physical relevance of the window. The Kuramoto-to-KS phenomenological bridge has a regime-dependent weight limit: the discrete-to-continuum interface is a genuine boundary where the sign of the forcing-structure effect reverses.

The epistemic protocol (predefined kill criteria, observables, competing hypotheses register) was designed before simulations and preserved claim discipline when the data contradicted the initial hypothesis.

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