Floquet Theory of the Modulated LC Circuit

Parametric resonance in LC circuits is a century-old subject, yet its exact mathematical structure has never been fully exposed in book form. This volume closes that gap. The nondissipative LC circuit with harmonically varying capacitance is governed by an exact equation belonging to Ince's four-parameter Hill class — not the Mathieu approximation used in virtually all engineering treatments. This identification has far-reaching consequences. Instability occurs only at odd sub-harmonics of the natural frequency; every even resonance is provably, exactly stable. The reason is algebraic: the Ince coexistence polynomial has an integer root, forcing two independent periodic solutions to coexist and collapsing the even instability intervals to zero width — a theorem, not a numerical accident. Closed-form formulas for the widths and boundary curves of all surviving instability tongues are derived, confirmed against Cambi's 1948 exact solution. Every instability boundary is shown to be an exceptional point of degeneracy (EPD) of the monodromy matrix, explained by Krein's collision theory for symplectic systems. A universal entire-function expansion of the Hill discriminant is established. Finally, the EPD structure is applied to hypersensitive capacitance sensing: a square-root frequency-split response to small perturbations, with an explicit closed-form formula and experimental protocol. This self-contained treatment is aimed at mathematical physicists, applied mathematicians, and electrical engineers interested in rigorous spectral theory and its sensing applications.