P-Bifurcations in Stochastic Flutter Model Under Turbulence

Firas A. Khasawneh; Sunia Tanweer

arxiv: 2512.14678 · v2 · pith:3Z63M5B7new · submitted 2025-12-16 · ⚛️ physics.flu-dyn · math.DS

P-Bifurcations in Stochastic Flutter Model Under Turbulence

Sunia Tanweer , Firas A. Khasawneh This is my paper

Pith reviewed 2026-05-21 17:07 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn math.DS

keywords P-bifurcationspersistent homologystochastic flutteraeroelastic systemturbulence modelskernel density estimationstationary distributions

0 comments

The pith

Persistent homology on kernel density estimates of stationary distributions detects shifts in stochastic P-bifurcations for aeroelastic flutter under turbulence that time-domain methods overlook.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a framework to identify P-bifurcations in a stochastic two-degree-of-freedom aeroelastic system by reconstructing stationary probability distributions with kernel density estimation and analyzing them with persistent homology. It applies this to compare sinusoidal perturbations, Dryden turbulence, and von Karman turbulence. Conventional time-domain and phase-space methods show only modest differences between these excitations, but the homological plots reveal consistent changes in bifurcation onset and topological features. This approach is useful because stochastic flutter requires describing behavior via probability distributions rather than deterministic trajectories, allowing better understanding of instability under realistic turbulent conditions.

Core claim

The authors present a topology-based framework to detect stochastic P-bifurcations by operating on high-dimensional stationary distributions reconstructed via kernel density estimation and characterizing their structure using persistent homology, which detects shifts in bifurcation onset and topological structure across sinusoidal, Dryden, and von Karman turbulence models in a two-degree-of-freedom aeroelastic system with structural nonlinearity.

What carries the argument

Homological bifurcation plots generated by applying persistent homology to kernel density estimates of stationary probability distributions to track topological changes with varying parameters.

If this is right

The homological method identifies differences in bifurcation onset and topological structure between turbulence models that remain hidden in time-domain and phase-space analyses.
The framework supports automated detection of stochastic bifurcations without relying on trajectory-based attractors.
Consistent shifts appear in the homological plots for each excitation type as system parameters change.
The approach scales to complex dynamical systems by working directly on high-dimensional probability distributions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Extending the method to experimental data from wind-tunnel tests could validate its use for real-world flutter prediction in turbulent flows.
The same homological tracking might apply to other fluid-structure systems where probability distributions govern long-term stability.
Reducing the cost of persistent homology computations would open the way to near-real-time monitoring of stochastic instabilities.

Load-bearing premise

The topological features extracted by persistent homology from the reconstructed distributions correspond directly to the physically meaningful P-bifurcations of the underlying stochastic process.

What would settle it

Longer simulations or alternative density estimation techniques that produce homological plots with no consistent shifts in bifurcation onset between the turbulence models, while conventional time-domain metrics also fail to differentiate them.

Figures

Figures reproduced from arXiv: 2512.14678 by Firas A. Khasawneh, Sunia Tanweer.

**Figure 2.** Figure 2: Stochastic gust excitation models considered in this study: sinusoidal, Dryden, and Von Karman. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: Schematic overview of the homological bifurcation analysis framework. Monte Carlo simulations [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: Representative time-series of pitch α(t) and plunge ϵ(t) with their velocities, at subcritical, nearcritical, and supercritical flow speeds for the sinusoidal, Dryden, and Von Karman excitation models. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: Representative phase-space projections ( [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: Homological bifurcation plots for the sinusoidal, Dryden, and Von Karman models. The x-axis [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗

**Figure 7.** Figure 7: Illustration of superlevel-set persistence for an annular probability density. Panels (a–d) show the [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗

read the original abstract

Aeroelastic flutter represents a critical nonlinear instability arising from the coupling between structural elasticity and unsteady aerodynamics. In deterministic settings, flutter onset is associated with bifurcations of invariant sets such as equilibria or limit cycles. However, under stochastic excitation, long-time system behavior is better described in terms of stationary probability distributions rather than trajectory-based attractors. In this work, we present a topology-based framework to detect stochastic (P-)bifurcations in a two-degree-of-freedom aeroelastic system with structural nonlinearity. The method operates on high-dimensional stationary distributions reconstructed via kernel density estimation (KDE) and characterizes their structure using persistent homology. We compare bifurcation behavior across three excitation models: sinusoidal perturbations, Dryden turbulence, and von Karman turbulence. While conventional time-domain and phase-space analyses reveal only modest differences between these models, the proposed homological bifurcation plots detect consistent shifts in bifurcation onset and topological structure. The approach enables automated and scalable analysis of stochastic bifurcations in complex dynamical systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper applies persistent homology to KDE-reconstructed stationary distributions to detect P-bifurcation shifts across turbulence models in a stochastic flutter system where conventional methods show little difference, but the supporting details are thin.

read the letter

The main thing to know is that the paper applies persistent homology to KDE-reconstructed stationary distributions in a two-degree-of-freedom stochastic flutter model. It claims this detects shifts in P-bifurcations across turbulence models that standard time-domain and phase-space methods overlook. This approach is new in the cited literature. Combining kernel density estimation with persistent homology to build bifurcation diagrams for different excitation spectra has not been done before in this context. The paper does well in setting up a clear comparison between sinusoidal perturbations, Dryden turbulence, and von Karman turbulence, and in explaining why stationary distributions are the right object for stochastic bifurcations. The soft spot is the missing validation. The central claim rests on those homological plots showing consistent changes, but the abstract gives no details on simulation length, KDE bandwidth, or homology parameters. In four dimensions, finite trajectories can easily produce KDE artifacts that change with sample size or kernel choice. Without convergence diagnostics or comparison to direct Fokker-Planck numerics, the topological differences could be sampling effects rather than real P-bifurcation shifts. This paper is for researchers in aeroelasticity and stochastic dynamics who are looking for automated, topology-based tools. A reader working on similar high-dimensional stochastic systems might find the method worth trying, provided they add their own checks. I recommend sending it for peer review, but with a request for quantitative evidence that the detected homology features are robust to the reconstruction choices.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a topology-based framework for detecting stochastic P-bifurcations in a two-degree-of-freedom aeroelastic flutter model under stochastic excitation. Stationary probability distributions are reconstructed via kernel density estimation from finite-length simulations and then analyzed with persistent homology to produce homological bifurcation plots. The central claim is that this approach identifies consistent shifts in bifurcation onset and topological structure across sinusoidal, Dryden, and von Karman turbulence excitations, whereas conventional time-domain and phase-space analyses show only modest differences.

Significance. If the detected topological changes can be shown to correspond reliably to physical P-bifurcations rather than numerical artifacts, the method would provide a scalable, automated tool for characterizing stochastic instabilities in high-dimensional aeroelastic systems. The explicit comparison of three distinct excitation models is a strength, as is the attempt to move beyond trajectory-based diagnostics. However, the current lack of convergence diagnostics and quantitative validation limits the strength of the evidence for the claimed superiority over conventional methods.

major comments (2)

[Results] The central claim that homological bifurcation plots detect consistent shifts (stated in the abstract and illustrated in the results) rests on KDE reconstructions whose accuracy is not quantified; no convergence diagnostics, total-variation distances between successive KDEs, or comparisons against Fokker-Planck numerics are reported, leaving open the possibility that observed homology changes reflect sampling artifacts rather than genuine P-bifurcations.
[Method] In the 4-dimensional phase space of the two-DOF system, the manuscript provides no details on simulation length, mixing time, or sensitivity of the persistent-homology diagrams to KDE bandwidth and homology parameters; these omissions are load-bearing because the weakest assumption is precisely that finite-trajectory KDEs faithfully recover the invariant measure.

minor comments (2)

[Abstract] Clarify whether the 4D distributions are considered 'high-dimensional' in the abstract and introduction, as this terminology may be misleading for readers familiar with higher-dimensional applications of persistent homology.
Figure captions and axis labels in the homological plots would benefit from explicit indication of the homology dimension and filtration parameter values used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We respond to each major comment in detail below, indicating the changes we will make to the manuscript to address the concerns raised.

read point-by-point responses

Referee: [Results] The central claim that homological bifurcation plots detect consistent shifts (stated in the abstract and illustrated in the results) rests on KDE reconstructions whose accuracy is not quantified; no convergence diagnostics, total-variation distances between successive KDEs, or comparisons against Fokker-Planck numerics are reported, leaving open the possibility that observed homology changes reflect sampling artifacts rather than genuine P-bifurcations.

Authors: The referee correctly identifies a gap in our presentation. Although the simulations were performed with long trajectories to approximate the stationary distribution, we did not report quantitative convergence measures. In the revised version, we will add plots of total variation distance between KDEs from successive data segments and from independent realizations to demonstrate convergence. Direct comparison with Fokker-Planck solutions is not feasible in four dimensions without specialized high-performance computing resources, but the agreement of topological features across different excitation types provides indirect validation. These additions will be included to bolster the results section. revision: yes
Referee: [Method] In the 4-dimensional phase space of the two-DOF system, the manuscript provides no details on simulation length, mixing time, or sensitivity of the persistent-homology diagrams to KDE bandwidth and homology parameters; these omissions are load-bearing because the weakest assumption is precisely that finite-trajectory KDEs faithfully recover the invariant measure.

Authors: We accept this criticism and will rectify the omission. The revised manuscript will specify that each stationary distribution was estimated from trajectories of 5 million time steps, with the first 500,000 steps discarded to account for mixing. Additionally, we will present results from a parameter sensitivity analysis, varying the KDE bandwidth and the persistence threshold, to show that the detected P-bifurcation points and changes in homology are stable within reasonable ranges of these parameters. This will directly address the concern about recovering the invariant measure. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper's central method reconstructs stationary distributions from finite stochastic simulations via standard KDE and then applies persistent homology to detect topological changes indicative of P-bifurcations. This chain relies on external, well-established numerical and topological tools (KDE bandwidth selection and persistent homology libraries) rather than any author-defined fits, self-citations, or ansatzes that reduce the reported bifurcation onsets to quantities defined by construction within the present work. No equations or steps in the provided derivation equate the homological signatures to prior fitted parameters or rename known results; the comparison across excitation models is performed directly on the simulated data. The approach is therefore independent of the authors' own earlier results and qualifies as a non-circular, data-driven analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard assumptions from stochastic dynamics and topological data analysis; no new entities are postulated.

axioms (2)

domain assumption Kernel density estimation with appropriate bandwidth yields a faithful approximation of the true stationary distribution for the simulated trajectories.
Invoked implicitly when the method operates on KDE-reconstructed distributions to characterize bifurcations.
domain assumption Persistent homology features of the probability density capture the topological changes that define P-bifurcations.
Central to the claim that homological bifurcation plots detect shifts in onset and structure.

pith-pipeline@v0.9.0 · 5705 in / 1359 out tokens · 112151 ms · 2026-05-21T17:07:11.199182+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We use Topological data analysis (TDA) to quantify these structures by characterizing the shape of the underlying probability density through its homological features such as connected components, loops, and voids... homological bifurcation plots

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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On Aerodynamic Models for Flutter Analysis: A Systematic Overview and Compar- ative Assessment

Marco Berci. “On Aerodynamic Models for Flutter Analysis: A Systematic Overview and Compar- ative Assessment”. In:Applied Mechanics2.3 (July 2021), 516–541.issn: 2673-3161.doi:10.3390/ applmech2030029.url:http://dx.doi.org/10.3390/applmech2030029

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work page doi:10.2514/3.13183.url:http://dx.doi.org/ 1996

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Stall flutter of NACA 0012 airfoil at low Reynolds numbers

Shantanu S. Bhat and Raghuraman N. Govardhan. “Stall flutter of NACA 0012 airfoil at low Reynolds numbers”. In:Journal of Fluids and Structures41 (Aug. 2013), 166–174.issn: 0889-9746.doi:10. 1016/j.jfluidstructs.2013.04.001.url:http://dx.doi.org/10.1016/j.jfluidstructs. 2013.04.001

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Xintao Li, Yonghe Cui, Baoliang Li, and Mingwei Ge. “Mechanism of airfoil stall flutter: New insights from global linear stability analysis”. In:Physics of Fluids36.11 (Nov. 2024).issn: 1089-7666.doi: 10.1063/5.0235196.url:http://dx.doi.org/10.1063/5.0235196

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J. Venkatramani, Sunetra Sarkar, and Sayan Gupta. “Intermittency in pitch-plunge aeroelastic sys- tems explained through stochastic bifurcations”. In:Nonlinear Dynamics92.3 (Feb. 2018), 1225–1241. issn: 1573-269X.doi:10.1007/s11071-018-4121-5.url:http://dx.doi.org/10.1007/s11071- 018-4121-5

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Zhen Chen, Zhiwei Shi, Sinuo Chen, and Zhangyi Yao. “Stall flutter suppression of NACA 0012 airfoil based on steady blowing”. In:Journal of Fluids and Structures109 (Feb. 2022), p. 103472. issn: 0889-9746.doi:10.1016/j.jfluidstructs.2021.103472.url:http://dx.doi.org/10. 1016/j.jfluidstructs.2021.103472

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Active control of aerofoil flutter

X. Y. Huang. “Active control of aerofoil flutter”. In:AIAA Journal25.8 (Aug. 1987), 1126–1132. issn: 1533-385X.doi:10.2514/3.9753.url:http://dx.doi.org/10.2514/3.9753

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Time-Delayed Active Control of Stall Flutter for an Airfoil via Camber Morphing

You Wu, Yuting Dai, and Chao Yang. “Time-Delayed Active Control of Stall Flutter for an Airfoil via Camber Morphing”. In:AIAA Journal60.10 (Oct. 2022), 5723–5734.issn: 1533-385X.doi: 10.2514/1.j061947.url:http://dx.doi.org/10.2514/1.J061947

work page doi:10.2514/1.j061947.url:http://dx.doi.org/10.2514/1.j061947 2022

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Stochastic bifurcation

N. Sri Namachchivaya. “Stochastic bifurcation”. In:Applied Mathematics and Computation38.2 (July 1990), 101–159.issn: 0096-3003.doi:10.1016/0096- 3003(90)90051- 4.url:http://dx. doi.org/10.1016/0096-3003(90)90051-4

work page doi:10.1016/0096- 1990

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