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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Bounded Channel-Adaptive Spectral Learning for Forward-Consistent Inverse Flapping-Wing Aerodynamics

Haichuan Li

University of Turku

Aerodynamics Inverse Dynamics Inverse Problem

Open paper AskPDF arXiv

Flapping-wing vehicles regulate aerodynamic forces and moments through coordinated variations in stroke, deviation, and pitch motion. Because the resulting loads depend on both the instantaneous wing configuration and its preceding motion history, recovering suitable wing kinematics from a desired aerodynamic trajectory is a challenging inverse problem. Existing sequence models capture temporal dependencies, while spectral methods can exploit the periodic structure of flapping motion. However, unrestricted frequency-domain augmentation may interfere with temporal representations and produce inconsistent corrections across kinematic variables and prediction horizons. We propose the Bounded Channel-Adaptive Spectral Residual Gated Recurrent Unit (BCS-GRU), which retains recurrent temporal prediction as its primary representation and restricts spectral information to a controlled output-specific correction. We further introduce a causally aligned forward-consistency objective that evaluates predicted kinematics through a separately trained and frozen aerodynamic surrogate. Experiments under a unified episode-level protocol show that BCS-GRU improves inverse prediction over recurrent and adaptive-spectral baselines, with greater benefits at longer prediction horizons. Forward-consistent fine-tuning further improves surrogate-based aerodynamic consistency while maintaining mean kinematic accuracy. These results demonstrate that controlled spectral correction and forward-consistent learning provide an effective framework for history-aware inverse modelling of flapping-wing aerodynamics.

Abstract

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