Applications and dynamics of delay differential equations with threshold state-dependent delay
Bibliographic record
Abstract
Delay differential equations (DDEs) with state-dependent delays defined by threshold conditions are studied.Threshold delays are implicitly determined by history states of the system over a bounded time interval.While threshold delays enhance model realism, they also pose analytical challenges.I study three different systems modeled by threshold DDEs, providing tangible strategies for tackling the threshold delays analytically and numerically to study their dynamics and bifurcations.Specifically, one approach focuses on reformulating the threshold condition through differentiation and stabilization when necessary to solve initial value problems.Another approach involves approximating the threshold delay by discretizing the distributed delay interval to solve a boundary value problem.Additionally, the effect of different delay forms, whether as a discrete constant or a discrete state-dependent delay, is examined in relation to the dynamics of the threshold delay problem.In Chapter 2 we consider an operon model with three differential equations for gene expression, accounting for time delays in DNA transcription, mRNA translation, and also models cell growth.We develop numerical techniques to solve the equations as initial value problems, and to perform numerical continuation and bifurcation detection.This is done by differentiating the threshold condition and discretizing the threshold delay.The implementations allow us to use standard software packages that are unable to handle threshold-type delays directly.Through numerical exploration of the bifurcations and dynamics of this system, we find that the inclusion of threshold delays broadens the possible operon dynamics.Namely, repressible systems may exhibit multistability and higher order bifurcation structures, and inducible systems may have stable periodic solutions.In Chapter 3 we analyze a scalar DDE modeling a transport process with a non-zero completion time, where the transport speed depends on the system's state.In contrast to the previous chapter, this model's scalar nature makes an in-depth analysis feasible.We perform systematic analytical and numerical studies of the various combinations of positive or negative feedback in the monotone nonlinear response function and the transport velocity to identify invariant objects including steady states, periodic orbits and their bifurcations.We identify rich bifurcation structures including Bautin, Bogdanov-Takens, cusp, fold, homoclinic, and Hopf bifurcations whose existence depends on the ii
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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) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 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".