转折点计数差异作为相对论轨道混沌的诊断指标
Turning-Point Count Discrepancy as a Diagnostic of Relativistic Orbital Chaos
- School of Physics and Technology, University of Jinan（济南大学物理科学与技术学院）
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中文总结 AI 辅助
提出转折点计数差异指标，从单条轨迹诊断相对论哈密顿系统中的轨道混沌，无需邻近轨道或相空间划分，适用于粒子和光子，并在克尔及带电粒子等模型中验证其有效性。
中文摘要 AI 辅助
我们提出转折点计数差异指标（TPCD），用于在具有两个振荡自由度的相对论哈密顿系统中，从单条轨迹诊断轨道混沌。TPCD 测量一个转折事件计数相对于每个参考周期的平均速率的最大累积偏差，既不需要邻近轨道，也不需要相空间划分，并且适用于大质量粒子和光子。我们在明确的事件-相位假设下建立了其长期行为。具有精确事件-相位对应的刚性相位满足严格的单位差异界，而具有有界一次相位变形的可线性化正则环面满足有限的、依赖于轨道的界；两者都意味着归一化指标随记录增长而衰减至零。相反，扩散涨落机制产生布朗桥标度和有限的统计尺度。可积的克尔运动验证了该构造，从事件计数中恢复指定频率比，对于六个目标（包括一个无理数比）精度达到 $2.6\ imes10^{-5}$。在外部测试磁场中围绕克尔黑洞的带电粒子扫描中，TPCD 和快速李雅普诺夫指标在所有 80 条采样轨迹上一致。在 Schwarzschild--Melvin 光子模型中，一条有限时间 TPCD 升高但快速李雅普诺夫值较低的轨迹，在其指标在长时间积分中呈下降趋势后被识别为规则轨迹，表明有限时间值必须与其长期趋势一起解读。
英文摘要
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 $2.6\times10^{-5}$ 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.
