Floquet Theory of the LC Circuit with Modulated Capacitance

Parametric resonance – periodic variation of a system parameter driving exponential growth of oscillations – is among the most fundamental instabilities in physics and engineering. The nondissipative LC circuit with harmonically varying capacitance is one of its simplest realizations: the modulation renders the circuit equation a Hill equation with either bounded or exponentially growing solutions. Identifying the governing equation as a special case of Ince’s four-parameter Hill equation yields two main results. First, a sharp structural theorem: instability occurs only at the odd sub-harmonics of the natural frequency, while every even resonance is exactly stable at all modulation amplitudes. This selectivity, invisible to the Mathieu approximation, follows from Krein’s collision theory: at odd resonances the colliding Floquet multipliers carry opposite Krein signatures, opening a tongue; at even resonances the signatures agree and the tongue collapses. Second, closed-form formulas for the widths and boundary curves of all surviving tongues, derived by a continued-fraction and Magnus–Winkler method, confirmed against Cambi’s 1950 numerics and recovered via the Yakubovich–Starzhinskii series. The continued fraction also gives the Floquet exponent as an exact power series in the modulation amplitude with rational coefficients, and finite-product formulas for all Fourier coefficients of the periodic Floquet factor. The tongue boundaries consist entirely of exceptional points of degeneracy (EPD) of the monodromy matrix, enabling hypersensitive capacitance sensing: at an EPD a small perturbation splits the coincident frequencies by the square root of its size, diverging relative to linear sensing as it shrinks. A closed-form splitting formula is derived, and a work-point strategy shifting slightly into the stable zone keeps the scheme robust while preserving square-root sensitivity.