Higher Order Structure along Nilpotent Eigendirections in Planar Vector Fields
Abstract
We study a distinguished one dimensional restriction associated with planar vector fields whose linearization is a nonzero nilpotent matrix of index two. This restriction admits an intrinsic interpretation on the nilpotent eigendirection, while its scalar representation in Jordan coordinates satisfies a simple transformation law under admissible changes of Jordan chain. Its quadratic Taylor coefficient coincides with the intrinsic quadratic Bogdanov--Takens coefficient. % For rational restrictions, we show that the Taylor coefficient sequence satisfies a finite linear recurrence determined by the minimal denominator. Its degree defines a recurrence degree that is invariant under admissible changes of Jordan chain and gives the minimal order of an eventual homogeneous recurrence. Exact cancellations may therefore be detected directly from the restriction before nonlinear normal form transformations are performed. Applications to four planar models exhibit different recurrence degree and cancellation mechanisms, illustrating the complementarity between the distinguished restriction and smooth nilpotent normal form theory.
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BibTeX
@misc{albarrangarcia2026higher,
title = {{Higher Order Structure along Nilpotent Eigendirections in Planar Vector Fields}},
author = {Roberto Albarran-García and Martha Alvarez Ramírez and Marco Polo García-Rivera},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27780},
eprint = {2608.27780},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27780},
}
RIS
TY - UNPB
TI - Higher Order Structure along Nilpotent Eigendirections in Planar Vector Fields
AU - Roberto Albarran-García
AU - Martha Alvarez Ramírez
AU - Marco Polo García-Rivera
PY - 2026
DA - 2026/08/27
PB - arXiv
DO - 10.48550/arxiv.2608.27780
UR - https://arxiv.org/abs/2608.27780
AB - We study a distinguished one dimensional restriction associated with planar vector fields whose linearization is a nonzero nilpotent matrix of index two. This restriction admits an intrinsic interpretation on the nilpotent eigendirection, while its scalar representation in Jordan coordinates satisfies a simple transformation law under admissible changes of Jordan chain. Its quadratic Taylor coefficient coincides with the intrinsic quadratic Bogdanov--Takens coefficient. % For rational restrictions, we show that the Taylor coefficient sequence satisfies a finite linear recurrence determined by the minimal denominator. Its degree defines a recurrence degree that is invariant under admissible changes of Jordan chain and gives the minimal order of an eventual homogeneous recurrence. Exact cancellations may therefore be detected directly from the restriction before nonlinear normal form transformations are performed. Applications to four planar models exhibit different recurrence degree and cancellation mechanisms, illustrating the complementarity between the distinguished restriction and smooth nilpotent normal form theory.
ER -