Piecewise linear perturbations of a linear center
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Bibliographic record
Abstract
This paper is mainly devoted to the study of the limit cycles that can bifurcate from a linearcenter usinga piecewise linear perturbation in two zones. We consider the case when the two zones are separatedby a straight line $\Sigma$ and the singular point of the unperturbed system is in $\Sigma$. It isproved that the maximum number of limit cycles that can appear up to a seventh order perturbation isthree. Moreover this upper bound is reached. This result confirms that these systems have more limitcycles than it was expected. Finally, center and isochronicity problems are also studied in systemswhich include a first order perturbation. For the latter systems it is also proved that,when theperiod function, defined in the period annulus of the center, is not monotone, then it has at mostone critical period. Moreover this upper bound is also reached.
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 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