Bounding extrema over global attractors using polynomial optimisation
Bibliographic record
Abstract
Abstract We present a framework for bounding extreme values of quantities on global attractors of differential dynamical systems. A global attractor is the minimal set that attracts all bounded sets; it contains all forward-time limit points. Our approach uses (generalised) Lyapunov functions to find attracting sets, which must contain the global attractor, and the choice of Lyapunov function is optimised based on the quantity whose extreme value one aims to bound. We also present a non-global framework for bounding extrema over the minimal set that is attracting in a specified region of state space. If the dynamics are governed by ordinary differential equations, and the equations and quantities of interest are polynomial, then our methods can be implemented computationally using polynomial optimisation. In particular, we enforce nonnegativity of certain polynomial expressions by requiring them to be representable as sums of squares, leading to a convex optimisation problem that can be recast as a semidefinite program and solved computationally. This computer assistance lets one construct complicated polynomial Lyapunov functions. Computations are illustrated using three examples. The first is the chaotic Lorenz system, where we bound extreme values of various monomials of the coordinates over the global attractor. In the second example we bound extreme values over a chaotic saddle in a nine-mode truncation of fluid dynamics that displays long-lived chaotic transients. The third example has two locally stable limit cycles, each with its own basin of attraction, and we apply our non-global framework to construct bounds for one basin that do not apply to the other. For each example we compute Lyapunov functions of polynomial degrees up to at least eight. In cases where we can judge the sharpness of our bounds, they are sharp to at least three digits when the polynomial degree is at least four or six.
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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 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 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".