Carryover of a saddle-node bifurcation after transforming a parameter into a variable
Bibliographic record
Abstract
Abstract In this article, we introduce and study the carryover of a saddle-node bifurcation, a concept that describes how a saddle-node bifurcation of a dynamical system is carried over into an extended dynamical system obtained by transforming one of the parameters of the original system into a variable. We show that additional transversality and singularity conditions are needed to guarantee the carryover of a saddle-node bifurcation and provide a graphical methodology with a two-parameter bifurcation diagram to verify that such conditions are met. The results are applied to a gene activation model when the parameter describing the signal for activation is transformed into a variable, and to a cell cycle regulatory model when the parameter describing the cell mass is transformed into a variable. In both cases, we show that a saddle-node bifurcation carryover takes place.
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How this classification was reachedexpand
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 | 0.001 | 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 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".