KAM theory for some dissipative systems

The uploaded paper states an a‑posteriori KAM theorem for conformally symplectic maps (Theorem 1), defines the moving frame M(θ) = [DK | J−1DKN], the S(θ) block, and the non-degeneracy matrix det([⟨S⟩ ⟨SB0⟩+⟨Ã1⟩; (λ−1)Id ⟨Ã2⟩]) ≠ 0, and sketches a Newton-like iteration leading to quadratic error reduction and the estimates ‖Ke−K‖ ≤ C4 C2 δ−2τ ‖E‖ and |µe−µ| ≤ C5‖E‖, exactly the architecture used by the candidate solution (automatic reducibility, splitting into tangential/normal cohomology equations, averaged compatibility system, and a quasi-Newton correction) . The paper’s proof is presented as a sketch, explicitly referring to the detailed proof in prior work, but the steps and objects match the model’s Phase‑2 solution, including the dropping of the R(θ)W(θ) term to obtain a solvable model equation and a quadratic improvement under Diophantine small-divisor estimates and a shrinking-strip iteration . The attractor interpretation for |λ|<1 is also consistent (block λ Id in the reduced cocycle; normal contraction) . Minor issues in the model’s write-up include an algebraic slip in the proof of M’s invertibility (one needs to premultiply by DK^T J, using the isotropy K*Ω=0), and the lack of an explicit mention that isotropy is used to obtain automatic reducibility; these are standard and easily corrected. Overall, both are correct and substantially the same proof strategy, with the paper giving a faithful sketch and the model giving a standard, consistent elaboration.