Intensity---A Metric Approach to Quantifying Attractor Robustness in ODEs
Bibliographic record
Abstract
Although mathematical models do not fully match reality, robustness of dynamical objects to perturbation helps bridge from theoretical to real-world dynamical systems. Classical theories of structural stability and isolated invariant sets treat robustness of qualitative dynamics to sufficiently small errors. But they do not indicate just how large a perturbation can become before the qualitative behavior of our system changes fundamentally. Here we introduce a quantity, intensity of attraction, that measures the robustness of attractors in metric terms. Working in the setting of ordinary differential equations on $\mathbb{R}^n$, we consider robustness to vector field perturbations that are time dependent or independent. We define intensity in a control-theoretic framework, based on the magnitude of control needed to steer trajectories out of a domain of attraction. Our main result is that intensity also quantifies the robustness of an attractor to time-independent vector field perturbations; we prove this by connecting the reachable sets of control theory to isolating blocks of Conley theory. In addition to treating classical questions of robustness in a new metric framework, intensity of attraction offers a novel tool for resilience quantification in ecological applications. Unlike many measurements of resilience, intensity detects the strength of transient dynamics in a domain of attraction.
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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 | 0.002 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.000 | 0.003 |
| Scholarly communication | 0.001 | 0.004 |
| Open science | 0.001 | 0.002 |
| 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".