A period–doubling bifurcation with slow parametric variation and additive noise
Bibliographic record
Abstract
Slow sinusoidal modulation of a control parameter can maintain a low–period orbit into parameter regions where the low–period orbit is locally unstable, and a higherperiod orbit would normally occur. Whether or not a bifurcation to higher period becomes evident during the modulation depends on the competing effects of stabilization by the modulation and destabilization by inherent very low level system noise. A transition, often rapid, from a locally unstable period–1 orbit to period–2, for example, can be triggered by noise. The competing effects are examined here for a period–doubling bifurcation of a general unimodal map. A nested set of three matched asymptotic expansions (a triple–deck) is used to describe the combined period–1 and period–2 response. The resulting solution gives estimates of whether and where an apparent period–doubling bifurcation occurs. Typical period–1 stability boundaries are obtained that include the effect of the amplitude and frequency of the variation, the noise level in the system, and the allowable maximum threshold level of period–2 response.
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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 | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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".