# Floquet Fibre Geometry and Higher-Order Reduced Coordinates for Off-Manifold Transients near Nonlinear Aeroelastic Flutter

arXiv: https://arxiv.org/abs/2609.14674 · HTML: https://mlsanity.com/paper/2609.14674/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 40 references that are in the map.
- First posted on arXiv 13 Sep 2026.

## Builds on (most cited first, 5 of 5)

- [Data-Driven Modeling and Prediction of Non-Linearizable Dynamics via Spectral Submanifolds](https://mlsanity.com/paper/2201.04976/) (2022)
- [Data-driven reduced order models using invariant foliations, manifolds and autoencoders](https://mlsanity.com/paper/2206.12269/) (2022)
- [Data-Driven Nonlinear Model Reduction to Spectral Submanifolds via Oblique Projection](https://mlsanity.com/paper/2503.21895/) (2025)
- [Existence and uniqueness of global Koopman eigenfunctions for stable fixed points and periodic orbits](https://mlsanity.com/paper/1911.11996/) (2019)
- [Data-driven modelling of autonomous and forced dynamical systems](https://mlsanity.com/paper/2512.12432/) (2025)

## Abstract (arXiv)

Assigning reduced coordinates to states near an attracting limit cycle requires the correct invariant-fibre geometry. The classical first-order phase-isostable chart obtained from adjoint Floquet modes projects along the strong-stable quotient fibre, whereas a metric-orthogonal complement of the retained slow bundle generally does not. We prove locally that a chart satisfying the linearised semiconjugacy relation leaves an O(delta^2) invariance residual, while projection along a non-invariant complement generically leaves an O(delta) term. For a nonlinear aeroelastic limit cycle, the metric-normal and strong-stable directions differ by 48.5 to 71.7 degrees, and metric-normal perturbations contain first-order retained phase and slow-amplitude components. Replacing the metric normal by the strong-stable fibre changes the measured residual scaling from delta^1.01 to delta^1.87 without fitted parameters. We then test learned higher-order corrections whose linearisation is pinned to the adjoint-Floquet chart, whose symmetry is exact, and whose reduced flow is fixed. Although they reduce the registered fixed-normalisation latent residual, post-hoc amplitude recalibration and adjoint-Floquet-targeted future consistency move or reverse the ranking. Because the learned maps already share the baseline's first-order gauge and the future target is supplied by the baseline chart, these diagnostics establish neither an independent positive nor negative higher-order result. Correct first-order Floquet geometry is therefore necessary in this benchmark, while the additional predictive value of the learned correction remains unidentified by the available representation-dependent diagnostics.