Symmetry and the Buchanan-Lillo conjecture: A resolution of the mixed feedback case
Bibliographic record
Abstract
Buchanan and Lillo both conjectured that oscillatory solutions of the first-order delay differential equation with positive feedback x ′ ( t ) = p ( t ) x ( τ ( t ) ) , t ≥ 0 , where 0 ≤ p ( t ) ≤ 1 , 0 ≤ t − τ ( t ) ≤ 2.75 + ln 2 , t ∈ R , are asymptotic to a shifted multiple of a unique periodic solution. This special solution can also be described from the more general perspective of the mixed feedback case (sign-changing p ), thanks to its symmetry (antiperiodicity). The analogue of this conjecture for negative feedback, p ( t ) ≤ 0 , was resolved by Lillo, and the mixed feedback analog was recently set as an open problem. In this paper, we resolve the case of mixed feedback, obtaining results in support of the conjecture of Buchanan and Lillo, underlining its link to the symmetry of the periodic solution. In particular, we obtain and describe the optimal estimates on the necessary delay for existence of periodic (more generally, nonvanishing) solutions, with respect to the period (oscillation speed). These apply to almost any first-order delay system, as we consider the general nonautonomous case, under minimal assumptions of measurability of the parameters. We furthermore discuss and elucidate the relations between the periodic and the nonautonomous case. • Resolution of an unresolved conjectures in delay equations on the unique- ness of the threshold of hyperbolicity. • Solution of the problem in the ”mixed feedback” case. • Development of a rich array of new auxiliary functions, results, and techniques. • Illustrating simulations for a delay equation with oscillatory coefficient sin(t).
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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 | 0.000 |
| 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".