A new formal series method and its application to the center problem in Z2-equivariant nilpotent vector fields
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Bibliographic record
Abstract
In this paper, center problems and bifurcation of limit cycles are considered for Z 2 -equivariant nilpotent vector fields. A new formal series method is developed for computing the focal values of the vector fields, which can be conveniently implemented using a computer algebraic system. As an application, the new method is applied to classify the centers for a class of quintic-order systems, which contains four conditions associated with a nilpotent singular point at the origin and two center conditions associated with an elementary center at infinity. Moreover, eight small-amplitude limit cycles in the neighborhood of the origin and nine large-amplitude limit cycles at infinity are obtained. This is the first time to investigate the synchronous bifurcation problem associated with a nilpotent singular point at the origin and a Hopf singular point at infinity.
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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.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 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 |
Machine scores (provisional)
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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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