LOCAL AND GLOBAL BIFURCATIONS TO LIMIT CYCLES IN A CLASS OF LIÉNARD EQUATION
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Bibliographic record
Abstract
In this paper, we study limit cycles in the Liénard equation: ẍ + f(x)ẋ + g(x) = 0 where f(x) is an even polynomial function with degree 2m, while g(x) is a third-degree, odd polynomial function. In phase space, the system has three fixed points, one saddle point at the origin and two linear centers which are symmetric about the origin. It is shown that the system can have 2m small (local) limit cycles in the vicinity of two focus points and several large (global) limit cycles enclosing all the small limit cycles. The method of normal forms is employed to prove the existence of the small limit cycles and numerical simulation is used to show the existence of large limit cycles.
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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) | 0.000 | 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