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

Puxue Tan

cs.CVSep 13, 2026

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

Authors: Puxue Tan

Organizations: William Wright Technology Centre (W-Tech) School of Mechanical & Aerospace Engineering Queen’s University Belfast 50 Malone Road, Belfast, BT9 5BS, United Kingdom

Abstract

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.

Figures & tables

Symbol	Meaning	Value
$\varepsilon$	static unbalance	$0.25$
$r^{2}$	squared radius of gyration	$0.5$
$a$	elastic-axis offset	$-0.1$
$\xi_{u}$	aerodynamic coefficient	$0.0113$
$\mu_{h},\mu_{\alpha}$	structural damping	$0.1$
$\xi_{h0}$	linear plunge stiffness	$0.2$

Table 1: Benchmark parameters and operating point. Structural and aerodynamic values are those of Table 1 of [ 22 ] , except

\xi_{h3}

, which takes one of the values of the source’s study of the cubic plunge coefficient in place of the Table-1 base value; the remaining quantities are computed from ( 14 ) and recorded with the computational provenance of each run.

Quantity	Value
neutral multiplier $\mu_{1}$	$1$
slow pair $\mu_{2,3}$	$0.382194\pm 0.773514\,i$
slow modulus $\|\mu_{2,3}\|$	$0.862784$
fast multiplier $\mu_{f}$	$0.140222$
fast exponent $\lambda_{f}$	$-0.343608$
whitened slow-frame condition number	$22.24258$

Table 2: Floquet structure at

V=7.8

and the geometry of the two candidate discarded directions. Angles and coefficients of

v_{\perp}

are measured in the metric ( 16 ) over

512

phases.

Map	Param.	Discards	Evidence role
metric-orthogonal	$0$	$v_{\perp}$	instructive contrast
adjoint-Floquet	$0$	$w_{f}$	zero-parameter baseline; established theory
Model A (shared quadratic prefactor)	$75$	$w_{f}$	pre-specified primary higher-order model
Model B (quadratic-form correction)	$210$	$w_{f}$	post-hoc corrective model

Table 3: The four coordinate maps and the role each plays. All share the metric, the phase origin, the amplitude normalisation, the reduced flow ( 25 ) and the evaluation protocol; they differ only in the reduction map. The adjoint-Floquet chart and Models A and B additionally share one on-orbit first-order normalisation, pinned by construction (Section 3.2 ); the fitted scalar of ( 29 ) used in Table 4 is an external sensitivity transformation applied afterwards, not a difference between the maps as constructed.

		Pre-specified	Recalibrated	Self-whitened
Endpoint	Map	(prospective)	(post hoc)	(post hoc)
one-step invariance	Model A (primary)	$0.8$	$1.084$ $[-0.118,\,0.096]$ , $7/20$	$1.135$
one-step invariance	Model B (corrective)	$0.55$	$0.819$ $[-0.044,\,0.225]$ , $13/20$	$0.893$
one-period rollout	Model A (primary)	$0.79$	$1.071$ $[-0.135,\,0.098]$ , $10/20$	$1.111$
one-period rollout	Model B (corrective)	$0.51$	$0.758$ $[-0.121,\,0.291]$ , $12/20$	$0.814$
five-period rollout	Model A (primary)	$1.68$	$2.267$ $[-0.501,\,-0.037]$ , $5/20$	$2.267$

Table 4: Held-out comparison of the learned maps against the zero-parameter adjoint-Floquet chart on

20

symmetry-distinct off-manifold units. Entries are the paired median error ratio of the learned map to the chart, so values below one favour the learned map, with the cluster-bootstrap

95\%

confidence interval (CI) on the median

\log_{10}

gain and the number of units favouring the learned map. The upper block reports the pre-specified endpoint alongside two post-hoc sensitivities of it: a validation-fitted global amplitude recalibration ( 29 ) and the self-whitened check, both disclosed post hoc. The lower block reports the adjoint-Floquet-targeted future-consistency diagnostic ( 30 ), which is reference asymmetric because the baseline chart supplies the target coordinates. None of the three post-hoc columns is a more canonical measurement of the pre-specified quantity, and none is designated the primary control; Section 5 states what they jointly do and do not identify.

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