发表机构
SOBIN Institute LLC（SOBIN研究所有限公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出用初始状态精度资源$B_N$量化预测非线性系统所需精度，并应用于准周期强迫逻辑映射，发现奇怪非混沌吸引子介于环面与混沌之间，具有对数型精度成本及长激活时间范围。
AI 中文摘要
预测一个非线性系统在有限时间范围内达到指定精度需要多少初始状态精度？我们通过一次性资源$B_N$来表述这个逆预测问题，$B_N$定义为在特定局部感知架构下，初始状态所需的二进制细化比特数。在常见的尺度可分离误差增长机制中，改变固定容差仅使$B_N$产生一个与$N$无关的偏移。一个可精确求解的圆/切比雪夫映射提供了混沌校准$B_N=N\log_2 m+O(1)$。然后我们将该框架应用于仅在相位上存在初始不确定性的准周期强迫逻辑映射。在解析的时间范围内，代表性的光滑环面、奇怪非混沌吸引子（SNA）和混沌机制表现出三级精度资源层级：分别是有界增长、对数型增长和线性型增长。对于SNA，代数相位敏感性$G_N^{\mathrm{op}}\sim N^\mu$意味着允许的初始不确定性量级为$N^{-\mu}$，因此精度成本与$\log N$成正比。相位网格细化表明，采样的最坏情况最大值随相位分辨率的增加在数值上保持稳定。在环面-SNA分形化点附近，基于导数的相位敏感性已经可以增长，而有限扰动在超过$10^6$次迭代中仍保持在固定操作容差以下，揭示了异常长的激活时间范围。因此，SNA可以占据光滑环面和普通混沌之间的中间有限精度资源类别，而其操作表现可以在比基于导数的敏感性长得多的时间尺度上发生。
英文摘要
How much initial-state precision is required to predict a nonlinear system to a prescribed accuracy over a finite horizon? We formulate this inverse prediction problem through a one-shot resource $B_N$, defined as the number of binary refinement bits required in the initial state by a specified local sensing architecture. Within a common scale-separable error-growth regime, changing a fixed tolerance changes $B_N$ only by an $N$-independent offset. An exactly solvable circle/Chebyshev map provides the chaotic calibration $B_N=N\log_2 m+O(1)$. We then apply the framework to the quasiperiodically forced logistic map under phase-only initial uncertainty. Over the resolved horizons, representative smooth-torus, strange-nonchaotic-attractor (SNA), and chaotic regimes exhibit a three-level precision-resource hierarchy: bounded, logarithmic-like, and linear-like growth, respectively. For the SNA, algebraic phase sensitivity $G_N^{\mathrm{op}}\sim N^μ$ implies an admissible initial uncertainty of order $N^{-μ}$, and therefore a precision cost proportional to $\log N$. Phase-grid refinement shows that the sampled worst-case maxima are numerically stable with increasing phase resolution. Near the torus--SNA fractalization point, derivative-based phase sensitivity can already grow while finite perturbations remain below a fixed operational tolerance for more than $10^6$ iterations, revealing exceptionally long activation horizons. Thus an SNA can occupy an intermediate finite-precision resource class between a smooth torus and ordinary chaos, while its operational manifestation can occur on a much longer timescale than its derivative-based sensitivity.
Comments17 pages, 12 figures
