Parametric Instability in Discrete Models of Spatiotemporally Modulated Materials

The paper proves (by perturbation plus direct Floquet computation) that UMFs occur at sums of two unmodulated natural frequencies divided by a positive integer, gives exact phase-selection rules for n=2, shows difference-combination resonances are not UMFs, identifies the spectral range 0<Ωm<2√(1+4Kc), and demonstrates zero-threshold instability at β=1 (undamped) with finite thresholds under damping; it also observes that higher-order (β≥2) tongues are extremely narrow for small Km and produce stability windows away from exact commensurability. The model solution reproduces these conclusions via a Hill–Fourier block reduction and two-mode multiple-scales, adds clean asymptotic scaling ρ=1+ΓβKm^β at exact resonance and an explicit small-ζ threshold formula. Aside from a minor wording slip in the model’s outline (“exponentially small width” vs. the correct O(Km^β) width), there is substantive agreement. The approaches differ (paper: perturbation + numerics; model: Floquet–Hill + slow-flow), but the results align.