用延迟坐标下的不变测度可唯一识别系统动态
Invariant Measures in Time-Delay Coordinates for Unique Dynamical System Identification
- 在时间延迟坐标下构建不变测度，实现动力学唯一识别
- 结合多延迟帧与不同可观测量，消除剩余不确定性
- 适用于混沌或噪声系统，适合物理建模场景
当点轨迹分析因混沌或噪声而不可行时，不变测度常用于分析物理系统，但无法唯一确定底层动力学。本文第一项成果表明，与状态坐标（如 $[x(t), y(t), z(t)]$）中的不变测度不同，时间延迟坐标（如 $[x(t), x(t-τ), dots, x(t-(m-1)τ)]$）中的不变测度可将系统动力学唯一确定至拓扑共轭意义。第二项成果通过组合多个具有不同可观测量的延迟帧构造的不变测度，在满足合适初值条件下，彻底消除残余歧义。这些理论保证依赖于信息丰富的可观测量和适当的延迟参数 ($m,τ$)，在某些场景下可能受限。我们通过一系列物理实例验证了理论，展示了延迟坐标中不变测度在实际系统识别中的鲁棒性。
原文摘要 · Abstract (English)
While invariant measures are widely employed to analyze physical systems when a direct study of pointwise trajectories is intractable, e.g., due to chaos or noise, they cannot uniquely identify the underlying dynamics. Our first result shows that, in contrast to invariant measures in state coordinates, e.g., $[x(t), y(t), z(t)]$, the invariant measure expressed in time-delay coordinates, e.g., $[x(t), x(t-τ),\ldots, x(t-(m-1)τ)]$, can identify the dynamics up to a topological conjugacy. Our second result resolves the remaining ambiguity: by combining invariant measures constructed from multiple delay frames with distinct observables, the system is uniquely identifiable, provided that a suitable initial condition is satisfied. These guarantees require informative observables and appropriate delay parameters ($m,τ$), which can be limiting in certain settings. We support our theoretical contributions through a series of physical examples demonstrating how invariant measures expressed in delay-coordinates can be used to perform robust system identification in practice.
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