Where Chaos Pauses: Sliding-Window Frequency Ratios Reveal Transient Resonances in Relativistic Orbits
Abstract
A long-time Poincaré section flattens a chaotic orbit into one static blur, erasing the order in which the orbit visits its own local structures. We introduce the sliding-window frequency ratio (SWFR), built from nothing but radial and polar turning events, to recover that order along individual relativistic orbits. Each window spans a fixed number of radial cycles and simply counts the polar events inside it; because event times are kept, any candidate interval can be sliced out and replotted as its own Poincaré section. No spectrum, no reference center, no basis functions. Integrable Kerr benchmarks recover the known frequency ratios, and for regular motion with uniformly bounded count deviations the counting-error bound falls off as O(W−1). The same counts then pay off in chaos. In magnetized Kerr spacetime a single charged orbit dwells near 3/5 and later near 4/7, and exactly those intervals open into fivefold and sevenfold section structures; a second orbit dwells near 1/2 with two lobes. In Schwarzschild--Melvin spacetime a photon holds 4/5 across a fivefold pattern. In every case the section covers exactly the interval SWFR selected: integer counts and phase-space geometry agree that a globally chaotic orbit is paying a temporary visit to a resonance.
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BibTeX
@misc{cao2026where,
title = {{Where Chaos Pauses: Sliding-Window Frequency Ratios Reveal Transient Resonances in Relativistic Orbits}},
author = {Wenfu Cao and Ying Wang and Hongsheng Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01604},
eprint = {2610.01604},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01604},
}
RIS
TY - UNPB
TI - Where Chaos Pauses: Sliding-Window Frequency Ratios Reveal Transient Resonances in Relativistic Orbits
AU - Wenfu Cao
AU - Ying Wang
AU - Hongsheng Zhang
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01604
UR - https://arxiv.org/abs/2610.01604
AB - A long-time Poincaré section flattens a chaotic orbit into one static blur, erasing the order in which the orbit visits its own local structures. We introduce the sliding-window frequency ratio (SWFR), built from nothing but radial and polar turning events, to recover that order along individual relativistic orbits. Each window spans a fixed number of radial cycles and simply counts the polar events inside it; because event times are kept, any candidate interval can be sliced out and replotted as its own Poincaré section. No spectrum, no reference center, no basis functions. Integrable Kerr benchmarks recover the known frequency ratios, and for regular motion with uniformly bounded count deviations the counting-error bound falls off as O(W^−1). The same counts then pay off in chaos. In magnetized Kerr spacetime a single charged orbit dwells near 3/5 and later near 4/7, and exactly those intervals open into fivefold and sevenfold section structures; a second orbit dwells near 1/2 with two lobes. In Schwarzschild--Melvin spacetime a photon holds 4/5 across a fivefold pattern. In every case the section covers exactly the interval SWFR selected: integer counts and phase-space geometry agree that a globally chaotic orbit is paying a temporary visit to a resonance.
ER -