The entry-exit function and nilpotent corrections in multidimensional slow-fast systems

Abstract

We study multidimensional entry-exit phenomena in slow-fast systems whose fast linearization contains a nilpotent component. We show that the classical spectral balance still determines the exit point as ε → 0, however the nilpotent component induces an algebraic amplification invisible to the spectrum, producing a ε|logε|-shift of the exit point for 0 <ε ≪ 1. We further extend the analysis to systems with several spectral branches and show that mode dominance follows a hierarchical, lexicographic selection principle: accumulated spectral growth determines the leading order, while the nilpotent component resolves degeneracies between spectrally equivalent modes.

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BibTeX RIS

BibTeX

@misc{borsotti2026entryexit,
  title = {{The entry-exit function and nilpotent corrections in multidimensional slow-fast systems}},
  author = {Jacopo Borsotti},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15670},
  eprint = {2610.15670},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15670},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - The entry-exit function and nilpotent corrections in multidimensional slow-fast systems
AU  - Jacopo Borsotti
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15670
UR  - https://arxiv.org/abs/2610.15670
AB  - We study multidimensional entry-exit phenomena in slow-fast systems whose fast linearization contains a nilpotent component. We show that the classical spectral balance still determines the exit point as ε → 0, however the nilpotent component induces an algebraic amplification invisible to the spectrum, producing a ε|logε|-shift of the exit point for 0 <ε ≪ 1. We further extend the analysis to systems with several spectral branches and show that mode dominance follows a hierarchical, lexicographic selection principle: accumulated spectral growth determines the leading order, while the nilpotent component resolves degeneracies between spectrally equivalent modes.
ER  -

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