Poincaré Sections by Inverse Cubic Hermite Interpolation
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Bibliographic record
Abstract
A Poincaré section is a trace in a plane, say q=0, of an orbiting trajectory coming up out of the plane. Periodic orbits show up as a single point in such a plane, which the orbit passes through repeatedly. Quasiperiodic orbits show up as closed curves, indicating that the orbits take place in a torus. Chaotic orbits have more complicated behaviour. If the trajectory is computed by a discrete numerical method such as the second-order, symplectic, Størmer–Verlet method (also called the leapfrog method), the points on the orbit will not usually hit the transecting plane exactly. Instead, there will be a point with q < 0 and then the next point will have q > 0. One wants to interpolate between those two points to find just where the continuous trajectory cuts the section. For a second-order method like the leapfrog method, cubic Hermite interpolation is accurate enough and frequently used. But a simpler method exists, namely inverse cubic Hermite interpolation, which will be explained in this paper. We demonstrate on the Henon–Heiles model. We also demonstrate use of the Maple Compiler to make the simulations faster, and plots[pointplot] to make the plotting faster.
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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) | 0.000 | 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