标题： 时间延迟坐标中的不变测度用于唯一动力系统识别 显示英文标题

标题： Invariant Measures in Time-Delay Coordinates for Unique Dynamical System Identification

摘要： 不变测度被广泛用于比较混沌动力系统，因为它们对噪声数据、不确定的初始条件和不规则采样具有鲁棒性。 然而，具有不同瞬态动力学的大类系统仍可能表现出相同的渐近统计行为，这在仅使用不变测度进行系统识别时会带来挑战。 受Takens的开创性嵌入理论的启发，我们提出在时间延迟坐标中研究不变测度，这些测度对底层动力学表现出更高的敏感性。 我们的第一个结果表明，时间延迟坐标中的单个不变测度可用于执行系统识别，直至拓扑共轭。 这一结果已经超越了原始状态坐标中不变测度的能力。 继续探索延迟坐标的强大功能，我们通过展示使用从不同可观测量构造的时间延迟坐标中的额外不变测度可以实现唯一的系统识别，从而消除了共轭关系中的所有歧义。 我们的发现提高了不变测度在系统识别中的有效性，并拓宽了测度论方法在建模动力系统中的应用范围。 显示英文摘要

摘要： Invariant measures are widely used to compare chaotic dynamical systems, as they offer robustness to noisy data, uncertain initial conditions, and irregular sampling. However, large classes of systems with distinct transient dynamics can still exhibit the same asymptotic statistical behavior, which poses challenges when invariant measures alone are used to perform system identification. Motivated by Takens' seminal embedding theory, we propose studying invariant measures in time-delay coordinates, which exhibit enhanced sensitivity to the underlying dynamics. Our first result demonstrates that a single invariant measure in time-delay coordinates can be used to perform system identification up to a topological conjugacy. This result already surpasses the capabilities of invariant measures in the original state coordinate. Continuing to explore the power of delay-coordinates, we eliminate all ambiguity from the conjugacy relation by showing that unique system identification can be achieved using additional invariant measures in time-delay coordinates constructed from different observables. Our findings improve the effectiveness of invariant measures in system identification and broaden the scope of measure-theoretic approaches to modeling dynamical systems.