{"id":"77163274-28e6-496c-9d9f-4d1f4539a6a1","arxiv_id":"2506.21275","paper_version":4,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"A diffusion-map and Gaussian-process pipeline reconstructs Hopf, pitchfork, and Neimark-Sacker bifurcation diagrams of 2D Navier-Stokes flows from simulation snapshots.","lead":"Scientists trained small machine-learning models on high-resolution fluid simulations, then used a standard continuation toolbox to draw the flows' full stability maps, including hidden unstable states. This makes bifurcation analysis much cheaper for fluid flows and demonstrates for the first time a secondary (Neimark-Sacker) bifurcation in a Navier-Stokes surrogate.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The surrogate is trained only on stable time-marching trajectories, yet the main results — unstable branches and critical Reynolds numbers — are computed in extrapolated latent regions the GPR never sees; the reported bifurcation points are unvalidated.","rationale":"The paper's pipeline is competently assembled and demonstrates a genuine capability: stable branches, limit cycles, Floquet multipliers for the primary cylinder bifurcation, and a visually convincing torus birth in the pinball are all obtained from the latent ROMs, and the lifted stable states match DNS well. The pinball result in particular is striking. Those successes, however, validate only the stable, attracting parts of the surrogate. The headline novelty — accurate continuation of unstable states and of a secondary Neimark-Sacker bifurcation with its unstable limit-cycle branch — requires the GPR to be accurate in a region that time-marching DNS never samples. GPR is an interpolator; its posterior mean in a data hole can be smooth but statistically hollow, and the posterior variance there is dominated by the prior. The 2-3% test-set errors in Fig. 18 are computed on randomly split snapshots: since a single DNS run contributes many serially correlated snapshots to both training and test subsets, the split is not a valid generalization test and gives no information about extrapolation. The a priori odd-symmetrization (Eq. 37) further weakens the channel case: the un-symmetrized surrogate (Fig. 10b) already shows a perturbed pitchfork, so applying the symmetry transform removes exactly the regression error that would otherwise expose the extrapolation problem. A clean version of the experiment would hold out entire Reynolds numbers or entire trajectories, propagate the GPR posterior variance through the continuation, and compare the unstable branches to full-order continuation. Until such a check is performed, the central accuracy claim is conditional. I therefore agree with the reader's weakest-assumption identification and keep the CONDITIONAL verdict unchanged.","tokens_in":33480,"tokens_out":6868,"duration_ms":84118,"concrete_test":"Compute the unstable branches directly from the full-order Navier-Stokes solver without the surrogate: for the channel, continue the symmetric steady solution a1 = 0 for Re in (44,70) using a Newton-Krylov or time-stepper-Arnoldi method on BASILISK; for the pinball, continue the unstable limit cycle for Re in (104.67,110) by a shooting method with a Poincare section. Project these full-order solutions onto the same DMs/POD latent coordinates and compare with the surrogate branches in Figs. 10(c) and 14. If the projected full-order branch deviates from the surrogate by more than the GPR 95% CI, the reported unstable branches and critical points are regression artifacts rather than properties of the Navier-Stokes flow.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim — that the learned ROMs enable 'accurate numerical bifurcation and stability analysis' — requires the GPR vector field g(a,Re) of Eq. (29) (or map of Eq. (30)) to be trustworthy not only on the stable attractors sampled by DNS (Section 2.2), but also at the unstable fixed points and limit cycles that continuation in Section 2.5 must pass through. That requirement is not met with any evidence. In the sudden-expansion case, a1 = 0 is an unstable steady state for Re > 43.12; DNS trajectories depart from it, so the training data (Fig. 16) has a hole near this state, and the GPR must interpolate the vector field through that hole. The odd-symmetrization in Eq. (37) then forces a pitchfork at whatever zero the antisymmetrized field has, so the reported Resb = 43.12 is not an independent discovery. In the pinball case, the branch of unstable limit cycles beyond Rens = 104.67 (Fig. 14) is never visited by the stable torus dynamics used for training; its continuation lives entirely in the GPR's extrapolative regime. The same holds, to a lesser degree, for the unstable base flow a = 0 in the cylinder case for Re > 48.48. The only full-order validation points shown (black dots in Figs. 9b, 10d, 14d-g) are stable DNS states, which do not constrain unstable branches. The reported 95% CIs (2-3%, Appendix A) are computed on a random 60/40 split of snapshots from the same attractors (a leaky split), so they do not measure uncertainty in the untrained unstable regions. Consequently, the critical Reynolds numbers 48.48, 43.12, and 104.67 — and the central 'accurate bifurcation analysis' claim — rest on an unvalidated regression extrapolation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a four-stage, fully data-driven pipeline ('embed-learn-lift') for reduced-order bifurcation and stability analysis of two-dimensional incompressible Navier-Stokes flows. In stage (a), either POD or parsimonious Diffusion Maps are used to embed high-dimensional DNS snapshots into low-dimensional latent coordinates; in stage (b), Gaussian Process regression learns the latent dynamics in continuous time (Eq. 29) or as a discrete map (Eq. 30), with the Reynolds number as an additional input; in stage (c), numerical continuation and stability analysis are performed in the latent space with MATCONT; in stage (d), solutions are lifted back to physical space via a closed-form POD map or a k-NN pre-image. The method is demonstrated on three benchmarks: the cylinder wake (Andronov-Hopf, Re_cr=48.48), the planar sudden-expansion channel (pitchfork, Re_sb=43.12), and the fluidic pinball (Neimark-Sacker, Re_ns=104.67), including computation of Floquet multipliers, limit-cycle periods, and unstable branches.","tokens_in":33924,"tokens_out":5897,"duration_ms":70350,"significance":"If the unstable branches and critical points were validated against full-order computations, the framework would be a useful extension of the equation-free paradigm, and the parsimonious DMs identification of a five-dimensional latent space for the pinball secondary instability is a genuinely interesting result. The paper is, however, primarily a demonstration: all three reported critical Reynolds numbers lie inside the training ranges and match known literature values, so the contribution is the construction and continuation of a surrogate rather than the prediction of previously unknown bifurcations. Strengths of the paper are its clear four-stage methodology, the use of an established continuation toolbox, the computation of Floquet multipliers in latent space, and the reconstruction of physical-space states through the pre-image problem. The main weakness is that all full-order validation points are stable attractors, while the headline claims concern accurate continuation of unstable branches and critical points that are not directly supported by the evidence presented.","major_comments":[{"comment":"The GPR surrogate is trained exclusively on stable time-marching DNS trajectories (Section 2.2 and Appendix A), but the continuation in Section 2.5 requires evaluating the learned vector field or map at unstable steady states and unstable limit cycles that are never visited by the training data. For the sudden-expansion case, a1=0 is unstable for Re>43.12, so the training data (Fig. 16) has a hole near that state and the GPR must interpolate the vector field through this hole. For the pinball, the unstable limit-cycle branch beyond Re_ns=104.67 (Fig. 14, dashed curves) is not observed in the stable torus trajectories used for training. The only full-order comparisons shown (black dots in Figs. 9b, 10d, 14d-g) are stable DNS states, which do not constrain the unstable branches. The reported critical Reynolds numbers and unstable branches are therefore properties of GPR interpolation/extrapolation, not established properties of the Navier-Stokes equations. A concrete remedy would be to validate the unstable branches and critical points against full-order Newton/Arnoldi or time-stepper computations at a few Reynolds numbers, or, at minimum, to report the GPR predictive variance along the continued branches and clearly label the unstable portions as unvalidated predictions.","section":"2.4-2.5, Figs. 9-14"},{"comment":"The odd-symmetrization in Eq. (37) makes a1=0 a fixed point by construction and enforces an exact pitchfork structure in the surrogate. Consequently, the reported Re_sb=43.12 is the point where the antisymmetrized GPR's linearization changes sign; it is not an independent detection of the Navier-Stokes pitchfork. The paper correctly notes in Section 2.5 that the transformation is applied to restore symmetry, but the abstract and conclusions present the pitchfork as correctly identified without stating that the bifurcation type has been imposed. Please state this assumption explicitly and support the Navier-Stokes-level claim with full-order validation or a clear comparison with the known literature value.","section":"Eq. (37), Section 4.3"},{"comment":"The reported 95% confidence intervals (2.03%, 3.04%, and 3.22% for cylinder, channel, and pinball) are computed on a random 60/40 split of snapshots drawn from the same attractors. This is a leaky split: training and test points lie on the same trajectories or attractors, so the uncertainty estimates measure interpolation accuracy on the attractors, not the accuracy of the vector field in the unstable latent regions used for continuation. A leave-one-Reynolds-number-out or another out-of-distribution evaluation would be far more informative for the bifurcation claims, and the current wording in Appendix A ('accurate and robust approximation ... across the explored parameter ranges') overstates what the test error demonstrates.","section":"Appendix A"},{"comment":"The abstract and conclusions claim that DMs-based ROMs outperform POD-ROMs, particularly for secondary instabilities such as the pinball Neimark-Sacker bifurcation, but no quantitative POD-GPR result for the pinball is presented. Section 4.4 states that POD-based ROMs fail, but the failure is not documented: it is not reported which latent dimensions were attempted, what the continuation produced, or what the reconstruction errors were. Since the outperformance claim is a central contribution, it needs a direct comparison, for example POD with d=2, d=5, and a few larger dimensions, showing the failure modes and the quantitative degradation relative to the DMs-GPR result.","section":"Section 4.4 and Conclusions"}],"minor_comments":[{"comment":"Section 3.1 states that 2000 temporal realizations are stored per Reynolds number for the cylinder, while Section 4.1 uses Nt=1000 for the POD snapshot matrix; the relationship between these two numbers (e.g., subsampling by a factor of two) is not explained.","section":"Section 4.1"},{"comment":"There is a typo in the Conclusions: 'numerial analysis' should read 'numerical analysis'.","section":"Conclusions"},{"comment":"The diffusion-map kernel scale is set to epsilon=median(d_ij) and the pre-image uses K=4 nearest neighbors, but no sensitivity analysis or robustness discussion is provided for these hyperparameters; a brief statement on how the reported results depend on them would strengthen the presentation.","section":"Section 2.3.2"},{"comment":"The text says 'the remaining Floquet multipliers are real and satisfy f1<1 and f2<1', but the table also contains the complex-conjugate pair f7 and f8; the wording should be clarified to say that the two real multipliers outside the critical pair are less than one in modulus, while f9=1 is the trivial multiplier.","section":"Table 1 and Section 4.4"},{"comment":"The term 'normal-forms' is used loosely for the GPR surrogates; the surrogates are not normal forms in the classical dynamical-systems sense. A brief definition of what 'surrogate normal-form' means in this paper would help avoid confusion.","section":"Introduction and Section 2.5"}],"recommendation":"major_revision","confidential_remarks":"The paper describes a reasonable demonstration of an embed-learn-lift pipeline, and the DMs-based identification of a five-dimensional latent space for the pinball is interesting. However, the published version should not claim validation of unstable branches or of 'accurate numerical bifurcation and stability analysis' without full-order checks of those branches. The circularity concern raised by the stress-test is real but, in my view, fixable: the authors can add targeted full-order validations, report out-of-distribution uncertainty, and temper the claims. I would not reject the paper on circularity grounds alone, but the current wording exceeds the evidence, and the POD comparison needs to be substantiated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper is a well-executed demonstration of a four-stage pipeline (embed-learn-lift) that combines POD or parsimonious diffusion maps, Gaussian process regression, and MATCONT continuation to produce bifurcation diagrams of 2D Navier-Stokes flows. The most novel piece is the continuation of unstable limit cycles through a Neimark-Sacker bifurcation in the fluidic pinball using a five-dimensional diffusion-map embedding—I haven't seen that done before. The paper is clearly written, the benchmarks are sensible, and the reconstructed high-dimensional fields match DNS well on stable branches.\n\nThe soft spots are real, and they center on validation. The GPR is trained only on stable attractor trajectories, yet the continuation traces unstable steady states and unstable limit cycles that the training data never visit. For the sudden-expansion case, the training set (Fig. 16) has a hole near a1=0 for Re>Resb, so the unstable symmetric branch is an interpolation through that hole. The pitchfork structure is then imposed by the antisymmetrization in Eq. (37), so Resb=43.12 is a property of the symmetrized fit, not an independent discovery. The cylinder Hopf (48.48) and pinball NS (104.67) values also fall inside the training Re ranges, so the ROM reproduces rather than predicts those transitions. The validation points in Figs. 9b, 10d, and 14d-g are all stable DNS states; they do not constrain the unstable branches. The reported 95% CIs come from a random 60/40 split of snapshots from the same attractors, which leaks temporal correlation and does not measure uncertainty in the untrained regions.\n\nThat said, the paper is honest about the symmetry trick and the limitations are addressable. What's missing is a proper validation of the unstable branches—e.g., against full-order Newton-Krylov continuation or direct stability calculations—and a time-series-aware data split. If the authors add that, the claims would be much stronger. As it stands, this is a promising proof-of-concept, not a validated tool for predicting unknown bifurcations.\n\nI'd send it to peer review—the methodology is interesting and the pinball result deserves scrutiny—but I'd tell the referee to focus on the extrapolation issue. I wouldn't cite it in my own work until the unstable branches are verified.","headline":"Promising demonstration of a data-driven bifurcation pipeline, but the headline bifurcation points are interpolations of training data, not validated predictions.","tokens_in":34495,"tokens_out":4953,"would_cite":false,"duration_ms":60747,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37M20","65P30","76D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that a data-driven embed-learn-lift pipeline can reproduce the bifurcation and stability structure of two-dimensional Navier-Stokes flows, including unstable branches and Neimark-Sacker transitions, from simulation data…","keywords":["diffusion maps","proper orthogonal decomposition","Gaussian process regression","Navier-Stokes equations","bifurcation analysis","Neimark-Sacker bifurcation","reduced-order models","Floquet multipliers"],"falsifier":"Run a Newton-Krylov continuation directly on the full discretized Navier-Stokes equations for the cylinder wake and compare the genuinely unstable steady and periodic states to those the POD-GPR ROM predicts when lifted to physical space; any disagreement larger than the reported roughly 12 percent maximum error at $\\mathrm{Re}=50$, or a Floquet-multiplier trend opposite to the ROM's, would show that the unstable branches are surrogate artifacts. For the pinball, withhold the post-bifurcation Reynolds values from training and test whether the DMs-GPR ROM still predicts $\\mathrm{Re}_{ns}\\approx 104.67$ with the same five coordinates.","tokens_in":33290,"feed_emoji":"🌊","tokens_out":6831,"duration_ms":76927,"temperature":0.7,"pith_summary":"The paper aims to establish that a fully data-driven pipeline can substitute for the intractable full-order bifurcation analysis of Navier-Stokes flows: embed high-dimensional snapshots into a low-dimensional latent manifold, learn the evolution equations there with a regression surrogate, run numerical continuation and stability analysis in those coordinates, and lift the results back to physical space. It argues that such surrogate normal-forms capture not just the stable attractors seen in simulations but also unstable steady states and unstable limit cycles, along with their Floquet multipliers, so that critical Reynolds numbers and full bifurcation diagrams become computable from data alone. The evidence is three benchmark two-dimensional flows of increasing dynamical complexity: the cylinder wake (Hopf), the sudden-expansion channel (pitchfork), and the fluidic pinball (Neimark-Sacker), with the pinball case showing that linear POD coordinates fail and Diffusion Maps succeed. A careful reader would care because the method offers a practical route to bring the full continuation toolkit of low-dimensional dynamics to high-dimensional computational-fluid-dynamics problems.","feed_headline":"Data-only surrogates trace unstable fluid branches and bifurcations","feed_subtitle":"A four-stage embed-learn-lift pipeline reproduces cylinder, expansion-channel, and pinball transitions from simulation snapshots.","key_machinery":"The load-bearing object is the four-stage embed-learn-lift loop: an encoder $\\Phi$ maps snapshots to latent coordinates (POD projection or Diffusion Maps with parsimonious eigenvector selection); a Gaussian Process regression model $g(a,\\mathrm{Re})$ learns either time derivatives or a discrete-time map in those coordinates; numerical continuation solves $g=0$ and the periodic boundary-value problem in latent space, yielding Floquet multipliers; and a pre-image solver (k-nearest-neighbor convex interpolation) lifts latent solutions back to velocity fields. For the pitchfork case, an additional symmetry-defeating step replaces $g$ by its odd part with respect to latent-state reversal, which restores the exact pitchfork structure that the learned surrogate alone only approximates. This machinery is what converts continuation and stability computations, normally too expensive for the full discretized Navier-Stokes equations, into low-dimensional operations.","core_discovery":"On the paper's own terms, the central discovery is that parsimonious Diffusion Maps coordinates, coupled with Gaussian Process regression, yield minimal-dimensional surrogate models on which the standard machinery of numerical bifurcation theory can be run faithfully. With two POD-GPR coordinates, the cylinder ROM places a supercritical Andronov-Hopf bifurcation at $\\mathrm{Re}_{cr}=48.48$ and continues stable limit cycles with the expected Floquet-multiplier structure; with one POD-GPR coordinate, an odd-symmetrized channel ROM places a pitchfork at $\\mathrm{Re}_{sb}=43.12$ and recovers two coexisting stable asymmetric solutions within a single model; with five Diffusion-Maps coordinates, the pinball ROM detects a Neimark-Sacker bifurcation at $\\mathrm{Re}_{ns}=104.67$, continues the unstable limit-cycle branch, and shows an invariant torus via Poincar\\'e sections. The authors take this as evidence that nonlinear manifold learning identifies the intrinsic dimension of the latent dynamics where POD fails, making demanding secondary-bifurcation analysis feasible.","pith_inferences":["An implication beyond the paper is that the five selected Diffusion-Maps coordinates for the pinball could serve as a data-driven intrinsic-dimension diagnostic: the parsimonious eigenvector-selection error identifies not only the dimension but also the need for nonlinear coordinates, which may be a cheap screening tool for secondary instabilities.","The paper does not test the unstable branches against independent full-order continuation; because the Gaussian Process surrogate is trained only on stable-attractor trajectories, its unstable-branch predictions are an extrapolation, so a validation against a matrix-free Newton-Krylov continuation of the full equations would strengthen or falsify the extrapolated parts of the diagrams.","The same loop should transfer to other dissipative partial differential equations with finite-dimensional inertial manifolds (for example convection or reacting flows), though the paper only demonstrates two-dimensional incompressible Navier-Stokes cases.","The symmetry-enforcement step is invoked only for the discrete reflection symmetry of the pitchfork; extending the idea to continuous symmetries such as translation or rotation is a natural next step that the authors do not demonstrate."],"forward_implications":["From simulation snapshots alone, critical Reynolds numbers for primary and secondary instabilities can be computed (48.48, 43.12, and 104.67 in the three benchmarks), avoiding direct Newton or time-stepper solves of the full discretized equations.","Unstable steady states and unstable limit cycles become accessible through latent-space continuation, so bifurcation diagrams can include branches that direct time-marching never visits.","Stability of periodic states is quantified by Floquet multipliers and periods computed from the latent ROM, not from expensive full-order linearizations.","Diffusion Maps is required for secondary bifurcations such as the fluidic pinball's Neimark-Sacker transition; POD-based ROMs do not reliably reproduce the torus birth.","For pitchfork transitions, symmetry must be imposed on the surrogate (odd-symmetry transformation) before continuation; otherwise the numerical diagram shows a perturbed pitchfork with a turning point instead of the true crossing."],"supporting_citations":[{"why":"Supplies the parsimonious Diffusion Maps methodology for learning latent fluid-flow dynamics and the DMs coordinates used in the pinball ROM.","marker":"[43]"},{"why":"Supplies the parsimonious selection of eigenfunctions, the mechanism that identifies exactly five DMs coordinates for the pinball.","marker":"[39]"},{"why":"Supplies the odd-symmetry model transformation that restores the pitchfork bifurcation in the sudden-expansion channel flow.","marker":"[78]"},{"why":"Supplies the numerical continuation and bifurcation toolkit used to trace steady states, limit cycles, and their stability in latent space.","marker":"[3, 4]"},{"why":"Supplies the k-nearest-neighbor convex interpolation pre-image solver that lifts latent-space solutions to physical velocity fields.","marker":"[87]"},{"why":"Supplies the cylinder-wake critical Reynolds number and spatial resolution reference against which the first benchmark is calibrated.","marker":"[97]"},{"why":"Supplies the pitchfork bifurcation characterization for symmetric expanding-channel flows that anchors the second benchmark.","marker":"[102]"},{"why":"Supplies the fluidic pinball Neimark-Sacker threshold and low-frequency modulation features used to validate the third benchmark.","marker":"[103]"}],"fun_headline_variants":["ML surrogates uncover bifurcations in fluid flows","Data-driven ROMs explore unstable fluid branches","Diffusion maps enable bifurcation analysis for Navier-Stokes","Surrogate learning predicts fluid instability transitions","Latent-space models chart flow bifurcations"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The surrogate is trained only on trajectories that settle onto stable attractors, but the continuation is then trusted in latent regions corresponding to unstable steady states and unstable limit cycles that the training data never visited; if the Gaussian-process vector field is wrong there, the reported unstable branches and critical Reynolds numbers are regression artifacts rather than properties of the Navier-Stokes flow.","fun_headline_variants_meta":{"raw":{"variants":["ML surrogates uncover bifurcations in fluid flows","Data-driven ROMs explore unstable fluid branches","Diffusion maps enable bifurcation analysis for Navier-Stokes","Surrogate learning predicts fluid instability transitions","Latent-space models chart flow bifurcations"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000224,"raw_usage":{"total_tokens":1533,"prompt_tokens":1091,"completion_tokens":442,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":707,"completion_tokens_details":{"reasoning_tokens":369}},"tokens_in":707,"tokens_out":442,"duration_ms":5828,"temperature":1.0,"reasoning_tokens":369,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:27:54.363813+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run a Newton-Krylov continuation directly on the full discretized Navier-Stokes equations for the cylinder wake and compare the genuinely unstable steady and periodic states to those the POD-GPR ROM predicts when lifted to physical space; any disagreement larger than the reported roughly 12 percent maximum error at $\\mathrm{Re}=50$, or a Floquet-multiplier trend opposite to the ROM's, would show that the unstable branches are surrogate artifacts. For the pinball, withhold the post-bifurcation Reynolds values from training and test whether the DMs-GPR ROM still predicts $\\mathrm{Re}_{ns}\\approx 104.67$ with the same five coordinates.","supporting_citations":[{"cited_title":"Della Pia, D","cited_arxiv_id":null,"evidence_quote":"Supplies the parsimonious Diffusion Maps methodology for learning latent fluid-flow dynamics and the DMs coordinates used in the pinball ROM."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the parsimonious selection of eigenfunctions, the mechanism that identifies exactly five DMs coordinates for the pinball."},{"cited_title":"Evangelou, D","cited_arxiv_id":null,"evidence_quote":"Supplies the odd-symmetry model transformation that restores the pitchfork bifurcation in the sudden-expansion channel flow."},{"cited_title":"Enabling equation-free modeling via diffusion maps.Journal of Dynamics and Differential Equations, 36(Suppl 1):415– 434, 2024","cited_arxiv_id":null,"evidence_quote":"Supplies the k-nearest-neighbor convex interpolation pre-image solver that lifts latent-space solutions to physical velocity fields."},{"cited_title":"Giannetti and P","cited_arxiv_id":null,"evidence_quote":"Supplies the cylinder-wake critical Reynolds number and spatial resolution reference against which the first benchmark is calibrated."},{"cited_title":"Battaglia, S","cited_arxiv_id":null,"evidence_quote":"Supplies the pitchfork bifurcation characterization for symmetric expanding-channel flows that anchors the second benchmark."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the fluidic pinball Neimark-Sacker threshold and low-frequency modulation features used to validate the third benchmark."}],"review_version":1}