PERIODIC SOLUTIONS OF THE CARSON-CAMBI EQUATION
Keywords:
frequency modulation, Carson-Cambi equation, coexistence problem, instability intervalsDOI:
https://doi.org/10.17654/0972096025010Abstract
We investigate the Carson-Cambi equation, a second-order linear differential equation arising in the modeling of frequency modulation circuits with a constant inductance and a periodically varying capacitance. We provide a comprehensive classification of nontrivial even and odd solutions with period or semi-period $\pi$. Using SturmLiouville theory and the framework of Hill and Ince equations, we analyze the spectral properties of the equation and describe the structure and distribution of its eigenvalues. We examine the coexistence of linearly independent periodic solutions, providing necessary and sufficient conditions via an associated polynomial criterion. In contrast to earlier studies limited by perturbative methods or numerical breakdown near $|a|=0.8$, we employ a nonperturbative framework based on self-adjoint operator theory and Prüfer angle analysis. This approach yields a Hochstadt-type estimate for instability intervals and supports accurate numerical computations even for modulation parameters approaching $|a|=1$, thus significantly extending the effective range of prior analyses.
Received: August 1, 2025
Accepted: September 1, 2025
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