发表机构
Concordia University; Cambridge University（康考迪亚大学; 剑桥大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出利用库普曼谱数据构造波包，在庞加莱截面上直接提取混沌系统的粗粒化输运结构，无需拟合回归映射，并在多个混沌系统中验证了其有效性。
AI 中文摘要
庞加莱截面将流替换为回归映射，但对于混沌系统，该映射通常仅能从采样的穿越中获知。我们证明，无需拟合映射，即可直接从库普曼谱数据读取粗粒化输运。保测EDMD保留了等距结构；riggedDMD随后近似谱测度并构造有限正则化波包。波包相位提供了有限分辨率的输运坐标；低模量标志着相位变得病态的奇异骨架。我们在Rössler系统、一个32模态Kuramoto--Sivashinsky伽辽金系统以及受迫Duffing振子上演示了这一思想。波包在从几乎一维曲线到明显厚集的截面上产生了粗粒化符号模型。它们的图组织了观测到的低周期轨道，并指导对其他轨道的定向搜索。在Duffing Regime II中，一个七区域规则解释了过滤后一步转移的91%–94%，而最低保留模量十分位中的失败发生率是整体率的5.08–5.20倍。波包不是库普曼本征函数，区域也不是精确的马尔可夫划分。这些计算共同表明，超出孤立本征对的谱信息可以直接从轨迹中揭示混沌输运。
英文摘要
A Poincaré section replaces a flow by a return map, but for a chaotic system this map is usually known only from sampled crossings. We show that coarse transport can be read directly from Koopman spectral data, without fitting the map. Measure-preserving EDMD retains the isometric structure; riggedDMD then approximates spectral measures and constructs finite regularized wave packets. Packet phase supplies a finite-resolution transport coordinate; low modulus marks a singular skeleton where the phase becomes ill-conditioned. We demonstrate the idea on the Rössler system, a 32-mode Kuramoto--Sivashinsky Galerkin system, and the forced Duffing oscillator. The packets yield coarse symbolic models on sections ranging from an almost one-dimensional curve to a visibly thick set. Their graphs organize observed low-period orbits and guide targeted searches for others. In Duffing Regime~II, a seven-region rule accounts for $91\%$--$94\%$ of filtered one-step transitions, while failures in the lowest retained modulus decile occur at $5.08$--$5.20$ times the overall rate. The packets are not Koopman eigenfunctions, nor are the regions exact Markov partitions. Together these computations show how spectral information beyond isolated eigenpairs can expose chaotic transport directly from trajectories.
