Investigating the consequences of global bifurcationsfor two-dimensional invariant manifolds of vector fields
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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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Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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