从流到映射：吸引子强度与有界噪声逃逸的采样定律
From Flows to Maps: Sampling Laws for Attractor Intensity and Bounded-Noise Escape
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中文总结 AI 辅助
该研究建立了连续时间吸引子强度与其时间 h 映射强度的采样对应关系，扩展了框架并通过实例验证，使离散观测的强度具有连续时间意义。
中文摘要 AI 辅助
吸引子的强度量化了吸引子在其 basin 内维持受控约束而不丢失的情况下，可承受的持续有界扰动的最大振幅。尽管强度已分别针对流和映射进行了表述，但其在时间采样下的行为仍未得到解决。我们建立了连续时间吸引子的强度 μ(A) 与其精确时间 h 映射的强度 μ_h(A) 之间的明确对应关系。对于 L-Lipschitz 向量场，有如下关系：μ(A)/(1+Lh) ≤ μ_h(A)/h ≤ μ(A)(e^{Lh}-1)/(Lh)，因此 μ_h(A)/h 趋近于 μ(A)。所得的一阶速率在一般情况下是尖锐的，而光滑标量逃逸几何可表现出二阶收敛。我们通过块强度的稳定性理论将该框架扩展到一步数值方法，并针对紧可逆非自治基上的吸引不变图，在驱动相位上获得了均匀采样收敛。对于有界支撑的随机扰动，归一化离散强度被确定为路径安全阈值；超过该阈值时，在明确的有限退出条件下会发生有限逃逸，而逃逸概率需要对噪声定律有额外假设。我们还证明了离散状态-法向边界映射收敛于控制极值可达集边界的归一化 Pontryagin 边界系统。精确标量基准、 grazing 恢复模型、平面 Duffing 逃逸、各向异性扰动、周期与准周期驱动以及转移算子计算均对该理论进行了说明。这些结果使从离散观测或模拟中估计的强度具有与采样无关的连续时间意义。
英文摘要
Intensity of attraction quantifies the largest amplitude of a persistent bounded disturbance that an attractor can withstand without loss of controlled confinement in its basin. Although intensity has been formulated separately for flows and maps, its behavior under temporal sampling has remained unresolved. We establish an explicit correspondence between the intensity $μ(A)$ of a continuous-time attractor and the intensity $μ_h(A)$ of its exact time-$h$ map. For an $L$-Lipschitz vector field, \[ \frac{μ(A)}{1+Lh} \leq \frac{μ_h(A)}{h} \leq μ(A)\frac{e^{Lh}-1}{Lh}, \] and hence $μ_h(A)/h\toμ(A)$. The resulting first-order rate is sharp in general, while smooth scalar escape geometries can exhibit second-order convergence. We extend the framework to one-step numerical methods through a stability theory for block intensity and to attracting invariant graphs over compact invertible nonautonomous bases, obtaining uniform sampling convergence over the forcing phase. For bounded-support random perturbations, normalized discrete intensity is identified with the pathwise safety threshold; above it, finite escape follows under an explicit finite-exit condition, while escape probabilities require additional assumptions on the noise law. We also show that the discrete state--normal boundary map converges to the normalized Pontryagin boundary system governing extremal reachable-set boundaries. Exact scalar benchmarks, a grazing resilience model, planar Duffing escape, anisotropic disturbances, periodic and quasiperiodic forcing, and transfer-operator computations illustrate the theory. These results give intensity estimated from discrete observations or simulations a sampling-independent continuous-time meaning.
