Turning-Point Count Discrepancy as a Diagnostic of Relativistic Orbital Chaos
Abstract
We propose the turning-point count-discrepancy indicator (TPCD) for diagnosing orbital chaos from a single trajectory in relativistic Hamiltonian systems with two oscillatory degrees of freedom. TPCD measures the largest cumulative departure of one turning-event count from its mean rate per reference cycle, requiring neither a neighboring orbit nor a phase-space partition and applying to both massive particles and photons. We establish its long-time behavior under explicit event--phase assumptions. Rigid phases with an exact event--phase correspondence obey a strict discrepancy bound of unity, and linearizable regular tori with bounded degree-one phase deformations obey a finite, orbit-dependent bound; both imply that the normalized indicator decays to zero as the record grows. A diffusive fluctuation mechanism instead yields a Brownian-bridge scaling and a finite statistical scale. Integrable Kerr motion validates the construction, recovering prescribed frequency ratios from event counts to within 2.6×10−5 for six targets, including an irrational ratio. In charged-particle scans around a Kerr black hole in an external test magnetic field, TPCD and the fast Lyapunov indicator agree for all 80 sampled trajectories. In the Schwarzschild--Melvin photon model, a trajectory with elevated finite-time TPCD but low fast-Lyapunov values is identified as regular once its indicator trends downward over an extended integration, showing that finite-time values must be read together with their long-time trend.
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BibTeX
@misc{cao2026turningpoint,
title = {{Turning-Point Count Discrepancy as a Diagnostic of Relativistic Orbital Chaos}},
author = {Wenfu Cao and Ying Wang and Hongsheng Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28914},
eprint = {2609.28914},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28914},
}
RIS
TY - UNPB
TI - Turning-Point Count Discrepancy as a Diagnostic of Relativistic Orbital Chaos
AU - Wenfu Cao
AU - Ying Wang
AU - Hongsheng Zhang
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.28914
UR - https://arxiv.org/abs/2609.28914
AB - We propose the turning-point count-discrepancy indicator (TPCD) for diagnosing orbital chaos from a single trajectory in relativistic Hamiltonian systems with two oscillatory degrees of freedom. TPCD measures the largest cumulative departure of one turning-event count from its mean rate per reference cycle, requiring neither a neighboring orbit nor a phase-space partition and applying to both massive particles and photons. We establish its long-time behavior under explicit event--phase assumptions. Rigid phases with an exact event--phase correspondence obey a strict discrepancy bound of unity, and linearizable regular tori with bounded degree-one phase deformations obey a finite, orbit-dependent bound; both imply that the normalized indicator decays to zero as the record grows. A diffusive fluctuation mechanism instead yields a Brownian-bridge scaling and a finite statistical scale. Integrable Kerr motion validates the construction, recovering prescribed frequency ratios from event counts to within 2.6×10^−5 for six targets, including an irrational ratio. In charged-particle scans around a Kerr black hole in an external test magnetic field, TPCD and the fast Lyapunov indicator agree for all 80 sampled trajectories. In the Schwarzschild--Melvin photon model, a trajectory with elevated finite-time TPCD but low fast-Lyapunov values is identified as regular once its indicator trends downward over an extended integration, showing that finite-time values must be read together with their long-time trend.
ER -