A geometric characterization of unstable blow-up solutions with computer-assisted proof.
Bibliographic record
Abstract
In this paper, blow-up solutions of autonomous ordinary differential equations (ODEs) which are unstable under perturbations of initial conditions are studied. Combining dynamical systems machinery (e.g. phase space compactifications, time-scale desingularizations of vector fields) with tools from computer-assisted proofs (e.g. rigorous integrators, parameterization method for invariant manifolds), these unstable blow-up solutions are obtained as trajectories on stable manifolds of hyperbolic (saddle) equilibria at infinity. In this process, important features are obtained: smooth dependence of blow-up times on initial points near blow-up, level set distribution of blow-up times, singular behavior of blow-up times on unstable blow-up solutions, organization of the phase space via separatrices (stable manifolds). In particular, we show that unstable blow-up solutions themselves, and solutions defined globally in time connected by those blow-up solutions can separate initial conditions into two regions where solution trajectories are either globally bounded or blow-up, no matter how the large initial points are.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".