Time-delay embedding is a powerful technique for reconstructing the dynamics of nonlinear systems. However, the reconstruction map is not always an embedding, a condition rarely verified in practice. When the reconstruction map is non-injective, multiple latent states may map to the same reconstructed state, leading to multi-valued $n$-step evolution. Consequently, the induced system no longer admits a deterministic closure, and the dispersion of future trajectories leads to ambiguity. In this work, we establish a measure-theoretic framework to quantify the ambiguity induced by multi-valued evolution and introduce intrinsic stochasticity to quantify the ambiguity over a finite horizon. For numerical implementation, we use the $k$-nearest-neighbor estimator to approximate intrinsic stochasticity under finite-resolution and finite-sampling settings. Numerical experiments on the synthetic and real-world datasets are consistent with the expectation: reconstructions closer to deterministic closure tend to produce lower scores, and deterministic predictors that take reconstructions with lower empirical closure scores as input are associated with lower rollout errors, suggesting that intrinsic stochasticity provides a new perspective for understanding failures of reconstruction and serves as a diagnostic for selecting reconstruction maps.

翻译：时间延迟嵌入是重构非线性系统动力学的一种强大技术。然而，重构映射并不总是嵌入，这一条件在实践中很少得到验证。当重构映射非单射时，多个潜状态可能映射到同一个重构状态，导致多值的$n$步演化。因此，诱导系统不再允许确定性封闭，未来轨迹的离散性会导致模糊性。在这项工作中，我们建立了一个测度论框架来量化多值演化所导致的模糊性，并引入内在随机性来量化有限时间范围内的模糊性。在数值实现中，我们使用$k$近邻估计量来近似有限分辨率和有限采样设置下的内在随机性。在合成和真实数据集上的数值实验与预期一致：更接近确定性封闭的重构倾向于产生更低的得分，而将具有较低经验封闭得分的重构作为输入的确定性预测器与更低的展开误差相关，这表明内在随机性为理解重构失败提供了新视角，并可作为选择重构映射的诊断工具。