Seven Limit Cycles Around a Focus Point in a Simple Three-Dimensional Quadratic Vector Field
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Bibliographic record
Abstract
In this paper, we show that a simple three-dimensional quadratic vector field can have at least seven small-amplitude limit cycles, bifurcating from a Hopf critical point. This result is surprisingly higher than the Bautin's result for quadratic planar vector fields which can only have three small-amplitude limit cycles bifurcating from an elementary focus or an elementary center. The methods used in this paper include computing focus values, and solving multivariate polynomial systems using modular regular chains. In order to obtain higher-order focus values for nonplanar dynamical systems, computationally efficient approaches combined with center manifold computation must be adopted. A recently developed explicit, recursive formula and Maple program for computing the normal form and center manifold of general n-dimensional systems is applied to compute the focus values of the three-dimensional vector field.
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Full frame distilled prediction
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.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| 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 |
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 it