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PreprintarXiv licencearXiv

Long-time stability of hierarchical point vortex configurations

  • Slim Ibrahim
  • Shengyi Shen
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Abstract

In this paper, we investigate the dynamics and long-time stability of hierarchical configurations in an N-point vortex system on ℝ2 and in simply connected domains, without the Cantor-set restriction inherent in KAM theory. Since the Hamiltonian perturbation lacks a standard power-series expansion in the usual normalized variables, we develop a modified Birkhoff normal form based on successive ratios of adjacent variables, tailored to the hierarchical structure. A key feature is the preservation of the resulting natural exponent structure throughout the normal form procedure. Consequently, for n sufficiently large, the hierarchical configuration persists for times of order at least ε−(n−2(N−1)).

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Cite

BibTeXRIS
BibTeX
@misc{ibrahim2026longtime,
  title = {{Long-time stability of hierarchical point vortex configurations}},
  author = {Slim Ibrahim and Shengyi Shen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23719},
  eprint = {2609.23719},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23719},
}
RIS
TY  - UNPB
TI  - Long-time stability of hierarchical point vortex configurations
AU  - Slim Ibrahim
AU  - Shengyi Shen
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23719
UR  - https://arxiv.org/abs/2609.23719
AB  - In this paper, we investigate the dynamics and long-time stability of hierarchical configurations in an N-point vortex system on ℝ^2 and in simply connected domains, without the Cantor-set restriction inherent in KAM theory. Since the Hamiltonian perturbation lacks a standard power-series expansion in the usual normalized variables, we develop a modified Birkhoff normal form based on successive ratios of adjacent variables, tailored to the hierarchical structure. A key feature is the preservation of the resulting natural exponent structure throughout the normal form procedure. Consequently, for n sufficiently large, the hierarchical configuration persists for times of order at least ε^−(n−2(N−1)).
ER  -