Dynamical invariants and inverse period-doubling cascades in multi-delay systems
Bibliographic record
Abstract
We investigate transitions to simple dynamics in first-order nonlinear differential equations with multiple delays. With a proper choice of parameters, a single delay can destabilize a fixed point. In contrast, multiple delays can both destabilize fixed points and promote high-dimensional chaos but also induce stabilization onto simpler dynamics. We show that the dynamics of these systems depend on the precise distribution of the delays. Narrow spacing between individual delays induces chaotic behavior, while a lower density of delays enables stable periodic or fixed point behavior. As the dynamics become simpler, the number of unstable roots of the characteristic equation around the fixed point decreases. In fact, the behavior of these roots exhibits an astonishing parallel with that of the Lyapunov exponents and the Kolmogorov-Sinai entropy for these multi-delay systems. A theoretical analysis shows how these roots move back toward stability as the number of delays increases. Our results are based on numerical determination of the Lyapunov spectrum for these multi-delay systems as well as on permutation entropy computations. Finally, we report how complexity reduction upon adding more delays can occur through an inverse period-doubling sequence.
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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 | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 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".