Parametric Resonance of a charged pendulum with suspension point oscillating between two vertical charged lines

The paper derives boundary surfaces via a Deprit–Hori normal form and lists ε-series that agree with the exact Mathieu boundaries for P1 (N=1,2,3) and for P2 (N=1,3) but reports a degenerate, constant boundary for P2 at N=2, α=(µ−4)/4 on both sheets, which contradicts the known nontrivial ε^2 splitting from Mathieu theory; see the paper’s formulas for P1 and P2 and the method description (k20k02=0) in Sections 3–4 and the explicit series for P1 (N=1,2,3) versus P2 (N=1,2,3) where N=2 is shown as coincident surfaces . The model’s solution reduces the linearized system exactly to the Mathieu equation, invokes Floquet theory and the characteristic values a_N(q), b_N(q), and recovers the correct two-sheet analytic boundaries (including the P2, N=2 splitting). The paper also lacks a rigorous justification that the truncated normal form’s conditions k20=0 or k02=0 coincide with the exact stability boundary, and it contains a minor sign swap in the conclusion’s stability inequalities .