Bibliographic record
Abstract
We present a method for constructing dynamical systems with robust, stable limit cycles in arbitrary dimensions. Our approach is based on a correspondence between dynamics in a class of differential equations and directed graphs on the n-dimensional hypercube (n-cube). When the directed graph contains a certain type of cycle, called a cyclic attractor, then a stable limit cycle solution of the differential equations exists. A novel method for constructing regulatory systems that we call minimal regulatory networks from directed graphs facilitates investigation of limit cycles in arbitrarily high dimensions. We identify two families of cyclic attractors that are present for all dimensions n ≥ 3: cyclic negative feedback and sequential disinhibition. For each, we obtain explicit representations for the differential equations in arbitrary dimension. We also provide a complete listing of minimal regulatory networks, a representative differential equation, and a bifurcation analysis for each cyclic attractor in dimensions 3–5. This work joins discrete concepts of symmetry and classification with analysis of differential equations useful for understanding dynamics in complex biological control networks.
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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 | 0.000 | 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 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".