Effect of parametric modulation on the stability of a periodic oscillator: a study with Airy functions

This work investigates the stability of a linear oscillator with a periodically varying stiffness composed of constant and linearly time-dependent segments. By combining Floquet theory with an analytical formulation in terms of Airy functions, the monodromy matrix is obtained in closed form and the characteristic multipliers that determine the stability regime are calculated. Unlike the classical literature on Hill or Mathieu systems, where the stiffness profile is assumed to switch instantaneously or vary sinusoidally, the present model explicitly incorporates finite transition times through linear ramps. This allows us to quantify how the duration of these transitions affects the onset of parametric resonance. The resulting stability map reveals alternating bands of stable and unstable regions reminiscent of Arnold tongues, and shows that the proportion of the cycle spent in the linear-ramp stage plays a decisive role in either promoting or suppressing instability. Overall, the study provides a compact analytical and numerical framework for assessing stability in periodically driven parametric systems of practical relevance in physics and engineering.