Investigating the consequences of global bifurcationsfor two-dimensional invariant manifolds of vector fields
Bibliographic record
Abstract
We consider a homoclinic bifurcation of a vector field in $\R^3$,where a one-dimensional unstable manifold of an equilibrium iscontained in the two-dimensional stable manifold of this sameequilibrium. How such one-dimensional connecting orbits arise is wellunderstood, and software packages exist to detect and follow them inparameters. In this paper we address an issue that it is far less well understood:how does the associated two-dimensional stable manifold changegeometrically during the given homoclinic bifurcation? This questioncan be answered with the help of advanced numerical techniques. Morespecifically, we compute two-dimensional manifolds, and theirone-dimensional intersection curves with a suitable cross-section, viathe numerical continuation of orbit segments as solutions of aboundary value problem. In this way, we are able to explain howhomoclinic bifurcations may lead to quite dramatic changes of theoverall dynamics. This is demonstrated with two examples. We firstconsider a Shilnikov bifurcation in a semiconductor laser model, andshow how the associated change of the two-dimensional stable manifoldresults in the creation of a new basin of attraction. We theninvestigate how the basins of the two symmetrically related attractingequilibria change to give rise to preturbulence in the firsthomoclinic explosion of the Lorenz system.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".