On the Stability of Smooth Branches of Periodic Solutions for Higher Order Perturbed Differential Equations

The paper proves stability criteria for smooth branches of periodic solutions via the Poincaré map’s Jacobian, using jets and Schur complements to factor the characteristic polynomial into P(ω; ε) and Q(λ; ε). The candidate solution reconstructs the same structure: (i) for m = n, reduce ∂zΠ to Id + A(ε) and decide stability from the jets of ε^{-`}A(ε), and (ii) for m < n, use the L-conjugation yielding a block form and Schur complements to define P and Q, whose jets control multipliers away from and near ω = 1, respectively. Minor methodological differences (the candidate invokes Kato/Implicit Function Theorem and a Neumann-series expansion where the paper uses Rouché and Schur-factorizations) are compatible and lead to the same conclusions. Key steps and hypotheses (H1–H4, simplicity on the unit circle and imaginary axis) match the paper’s statements and proofs, e.g., equations (7), (19)–(24) and Theorems A and B with conditions (a1)–(b2) .