Partial Breakdown of Integrability in Nearly Integrable Systems
Abstract
We study the persistence of integrable structures in real-analytic nearly integrable systems under perturbations that may destroy some, but not all, of these structures. Specifically, we establish upper bounds on the numbers of real-analytic first integrals -- quantities conserved by the motion -- and commuting vector fields -- infinitesimal generators of continuous symmetries -- that can persist under the perturbation. These bounds are expressed in terms of Melnikov-type functions, which quantify the breakdown of the corresponding integrable structures. Unlike classical Melnikov theory, our conditions occur even in the absence of zeros of the Melnikov functions, so they do not necessarily imply the existence of homoclinic tangles. We give a geometric interpretation of these conditions in the context of the Arnold diffusion problem, showing that they imply micro-diffusion. We also obtain a criterion for higher-dimensional nearly integrable systems in which the dimension of the span of the Melnikov vectors restricts the numbers of possible first integrals and commuting vector fields. This shows that the geometry of the Melnikov vectors provides a new mechanism for detecting the breakdown of integrability. We give several applications to integrable Hamiltonian systems subject to forcing and dissipation.
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BibTeX
@misc{gidea2026partial,
title = {{Partial Breakdown of Integrability in Nearly Integrable Systems}},
author = {Marian Gidea and Shoya Motonaga},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30354},
eprint = {2610.30354},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30354},
}
RIS
TY - UNPB
TI - Partial Breakdown of Integrability in Nearly Integrable Systems
AU - Marian Gidea
AU - Shoya Motonaga
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.30354
UR - https://arxiv.org/abs/2610.30354
AB - We study the persistence of integrable structures in real-analytic nearly integrable systems under perturbations that may destroy some, but not all, of these structures. Specifically, we establish upper bounds on the numbers of real-analytic first integrals -- quantities conserved by the motion -- and commuting vector fields -- infinitesimal generators of continuous symmetries -- that can persist under the perturbation. These bounds are expressed in terms of Melnikov-type functions, which quantify the breakdown of the corresponding integrable structures. Unlike classical Melnikov theory, our conditions occur even in the absence of zeros of the Melnikov functions, so they do not necessarily imply the existence of homoclinic tangles. We give a geometric interpretation of these conditions in the context of the Arnold diffusion problem, showing that they imply micro-diffusion. We also obtain a criterion for higher-dimensional nearly integrable systems in which the dimension of the span of the Melnikov vectors restricts the numbers of possible first integrals and commuting vector fields. This shows that the geometry of the Melnikov vectors provides a new mechanism for detecting the breakdown of integrability. We give several applications to integrable Hamiltonian systems subject to forcing and dissipation.
ER -