NONINTEGRABILITY OF TIME-PERIODIC PERTURBATIONS OF ANALYTICALLY INTEGRABLE SYSTEMS NEAR HOMO- AND HETEROCLINIC ORBITS

The paper proves Theorem 1.2 by: (i) rewriting the time-periodic system as the autonomous extension (1.8) under (A8), and reducing to a single-frequency complex subsystem (2.1) via Lemma 2.1; (ii) constructing the VE/AVE along a heteroclinic orbit, defining the Melnikov function using the bounded AVE solution ψ2; (iii) computing monodromy matrices around the endpoints with a triangular connection whose off-diagonal is the relevant Fourier coefficient M̂ℓ(c) of the Melnikov function; and (iv) invoking the Ayoul–Zung criterion (Theorem A.1) to conclude nonintegrability when M is nonconstant. See Theorem 1.2 and the reduction (1.8), (2.1) , the AVE/Melnikov setup , the VE asymptotics and bounded AVE solution , the monodromy calculation (Lemma 3.1) , and Theorem A.1 . The candidate solution follows the same strategy but makes a crucial technical error in the monodromy factor: it asserts T_+ multiplies w by e^{2π i}, which equals 1 and would not induce noncommutation; the paper’s correct local chart computation yields scaling by e^{2πℓν/λ1±} (and a triangular conjugation by B0 with off-diagonal M̂ℓ(c)), which is precisely what forces noncommutation when M̂ℓ(c) ≠ 0 . Consequently, the paper’s argument is correct, while the model’s writeup contains a substantive slip in this key step.