The Spectral Skeleton of Chaos: Koopman Wave Packets on Poincaré Sections
Abstract
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.
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BibTeX
@misc{bramburger2026spectral,
title = {{The Spectral Skeleton of Chaos: Koopman Wave Packets on Poincaré Sections}},
author = {Jason J. Bramburger and Shiloh-Sheray Gagnon and Rachel Morris and Matthew J. Colbrook},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06565},
eprint = {2609.06565},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06565},
}
RIS
TY - UNPB
TI - The Spectral Skeleton of Chaos: Koopman Wave Packets on Poincaré Sections
AU - Jason J. Bramburger
AU - Shiloh-Sheray Gagnon
AU - Rachel Morris
AU - Matthew J. Colbrook
PY - 2026
DA - 2026/09/06
PB - arXiv
DO - 10.48550/arxiv.2609.06565
UR - https://arxiv.org/abs/2609.06565
AB - 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.
ER -