Bounding extrema over global attractors using polynomial optimisation
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
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.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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) | 0.000 | 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 it