Bistability, bifurcations and chaos in the Mackey-Glass equation
Bibliographic record
Abstract
Numerical bifurcation analysis, and in particular two-parameter continuation, is used in consort with numerical simulation to reveal complicated dynamics in the Mackey-Glass equation for moderate values of the delay close to the onset of chaos. In particular a cusp bifurcation of periodic orbits and resulting branches of folds of periodic orbits effectively partition the parameter space into regions where different behaviours are seen. The cusp bifurcation leads directly to bistability between periodic orbits, and subsequently to bistability between a periodic orbit and a chaotic attractor. This leads to two different mechanisms by which the chaotic attractor is destroyed in a global bifurcation with a periodic orbit in either an interior crisis or a boundary crisis. In another part of parameter space a sequence of subcritical period-doublings is found to give rise to bistability between a periodic orbit and a chaotic attractor. Torus bifurcations, and a codimension-two fold-flip bifurcation are also identified, and Lyapunov exponent computations are used to determine chaotic regions and attractor dimension.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".