Winding Numbers in Discrete Dynamics: From Circle Maps and Fractals to Chaotic Poincaré Sections
Bibliographic record
Abstract
Winding numbers are key indices in the depiction, modelling, and testing of dynamical processes. They capture phase progression on closed curves and are robust for quasiperiodic dynamics, but their status for chaotic Poincaré sections is unclear. This study tests whether any non-trivial winding-type index can be extracted from chaotic Poincaré maps using three approaches: (i) phase-angle analysis, (ii) Kabsch optimal-rotation estimation, and (iii) local turning-angle averaging. To benchmark feasibility and error, we compare four systems: the standard circle map, the same circle map embedded on two planar fractal curves (Koch snowflake and Hilbert curve), a quasiperiodic Duffing–van der Pol (DVP) Poincaré map, and a chaotic DVP Poincaré map. For the quasiperiodic map, all methods yield consistent, accurate winding numbers. For the transitional systems (circle map and its fractal embeddings), indices remain non-trivial but more deviated. In stark contrast, chaotic Poincaré maps produce only trivial indices across all methods. These results indicate a crucial fact about the modelling of chaotic Poincaré maps. That is, although being fractal, they are not merely chaotic maps on fractal curves; rather, they reflect a tighter coupling of geometry and dynamics. Practically, the recoverability of a non-trivial winding index offers a simple diagnostic to distinguish quasiperiodicity from chaos in Poincaré data or corresponding models. The constructed chaotic-map-on-fractal systems also act as test-bed models that bridge ideal one-dimensional mappings and realistic two-dimensional Poincaré sections.
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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 | 0.001 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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".