当更多数据变得信息量更少：有限精度周期化与预测误差李雅普诺夫估计的崩溃
When More Data Become Less Informative: Finite-Precision Periodicization and Collapse of Forecast-Error Lyapunov Estimates
AI总结：
该研究发现增加有限精度混沌记录的长度会导致预测误差最大李雅普诺夫指数估计崩溃，独立重启可延迟该饱和，且与重现尺度高度相关。
AI中文摘要：
最大李雅普诺夫指数（LLE）用于量化指数敏感性，但数据驱动的估计常从有限精度轨迹中获取。本文表明，增加单条低精度混沌记录的长度最终会降低预测误差LLE估计的准确性。以r=4的逻辑斯蒂映射为例，NumPy float32可逐位复现ESP32单精度轨迹。在10000个随机float32初始条件下，每条轨迹均在迭代7612次前达到精确重现。对于一条长float32记录，当N=15000时，估计的LLE为0.6853；当N=20000时降至0.1827；当N=30000时约为0，这是因为精确的训练-测试历史达到饱和。当N=100000时，该长float32记录给出的LLE为0.0016，而独立重启的长度为100的轨迹给出的LLE为0.6917；匹配的float64对照组则保持在ln(2)=0.6931附近。该崩溃现象在28个代表性初始条件下均得到复现，且其发生与瞬态长度和数字周期设定的重现尺度高度相关（皮尔逊相关系数r=0.982）。因此，有限状态重现会将额外样本转化为重复的未来而非新的动力学信息，而独立重启能大幅延迟这种饱和现象。
英文摘要：
Largest Lyapunov exponents (LLEs) quantify exponential sensitivity, but data-driven estimates are often obtained from finite-precision trajectories. We show that increasing the length of a single reduced-precision chaotic record can eventually degrade a forecast-error LLE estimate. Using the logistic map at r=4, an ESP32 single-precision trajectory is reproduced bit-for-bit by NumPy float32. Across 10,000 random float32 initial conditions, every trajectory reaches an exact recurrence before iteration 7612. For one long float32 record, the estimated LLE changes from 0.6853 at N=15,000 to 0.1827 at N=20,000 and approximately zero at N=30,000 as exact train-test histories saturate. At N=100,000, the long float32 record gives 0.0016, whereas independently restarted length-100 trajectories give 0.6917; matched float64 controls remain near ln(2)=0.6931. The collapse is reproduced for 28 representative initial conditions, and its onset is strongly correlated with the recurrence scale set by transient length and digital period (Pearson r=0.982). Thus, finite-state recurrence can turn additional samples into duplicate futures rather than new dynamical information, while independent restarts substantially delay this saturation.
