Stability Analysis of Nondifferentiable Systems
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Bibliographic record
Abstract
ABSTRACT Differential equations with right‐hand side functions that are not everywhere differentiable are referred to as nondifferentiable systems. This paper introduces three novel methods to address stability issues in nondifferentiable systems. The first method extends the linearization method as it fails when the equilibrium is in a nondifferentiable region. We find that the stability of a piecewise differentiable system aligns with the behavior of its subsystems as long as the “distance” between these subsystems is sufficiently small. The second method is to examine the eigenvalues of the symmetric part of the Jacobian matrix in the vicinity of the equilibrium. This method applies to functions with even weaker regularity conditions, and does not require the eigenvalues to have a negative upper bound (or positive lower bound) over the domain. The third method establishes a connection between nondifferentiable systems and their approximate counterparts, revealing that their stability can be consistent under certain conditions. Additionally, we reaffirm the first two results via the approximation method. Examples are provided to illustrate the applications of our main results, including piecewise differentiable systems, general nondifferentiable systems, and realistic scenarios.
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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) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| 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 it