Bifurcation Diagrams and Moduli Spaces of Planar Quadratic Vector Fields with Invariant Lines of Total Multiplicity Four and Having Exactly Three Real Singularities at Infinity
Bibliographic record
Abstract
In this article we consider the class $${{\bf QSL}_{\bf4}^{\bf3s\boldsymbol\infty}}$$ of all real quadratic differential systems $${\frac{{\rm{d}}x}{{\rm{d}}t}=p(x,y),\frac{{\rm{d}}y}{{\rm{d}}t}=q(x,y)}$$ with gcd(p, q) = 1, having invariant lines of total multiplicity four and three real distinct infinite singularities. Firstly we construct compactified canonical forms for the class $${{\bf QSL}_{\bf4}^{\bf3s\boldsymbol\infty}}$$ so as to include limit points in the 12-dimensional parameter space of the set $${{\bf QSL}_{\bf4}^{\bf3s\boldsymbol\infty}}$$ . We next construct the bifurcation diagrams for these compactified canonical forms. These diagrams contain many repetitions of phase portraits and we show that these are due to many symmetries under the group action. To retain the essence of the dynamics we finally construct the moduli spaces under the action of the group of affine transformations and time homotheties and we place the phase portraits in these moduli spaces. The final diagrams retain only the necessary information to capture the dynamics under the motion in the parameter space as well as under the group action.
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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 | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".