An Adaptive Phase-Amplitude Reduction Framework Without O(ε) Constraints on Inputs

The paper derives an adaptive phase–amplitude reduction with no O(ε) constraints on input and provides an Appendix A bound showing the truncated isostables are O(ε) under bounded Ue and ṗ and fast decay κ = O(1/ε); see the statement “there are no O(ε) restrictions on the input U(t)” and the final Appendix claim after (A8) that no O(ε) constraints are imposed on Ue (main text and Appendix A) . The key inequality (A7) establishes max|ψk| = O(ε) for the neglected modes, leading to O(ε)+O(ψ^2) remainders in (A8) and O(ε)-accurate state reconstruction (19) . The candidate solution reaches the same conclusion but via a variation-of-constants argument and an explicit boundary-layer estimate, while the paper uses a differential-inequality bound; assumptions align (fast modes, ψ=O(√ε), bounded inputs), so both are correct.