Higher Order Structure along Nilpotent Eigendirections in Planar Vector Fields

[2608.27780] Higher Order Structure along Nilpotent Eigendirections in Planar Vector Fields Skip to main content Search Submit Donate Log in Mathematics > Dynamical Systems arXiv:2608.27780 (math) [Submitted on 27 Aug 2026] Title: Higher Order Structure along Nilpotent Eigendirections in Planar Vector Fields Authors: Roberto Albarran-García , Martha Alvarez Ramírez , Marco Polo García-Rivera View PDF HTML (experimental) 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. Subjects: Dynamical Systems (math.DS) ; Classical Analysis and ODEs (math.CA) MSC classes: 34C20, 34C23, 37G05, 37G10 Cite as: arXiv:2608.27780 [math.DS] (or arXiv:2608.27780v1 [math.DS] for this version) https://doi.org/10.48550/arXiv.2608.27780 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Martha Alvarez-Ramírez [ view email ] [v1] Thu, 27 Aug 2026 23:25:14 UTC (19 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.DS < prev | next > new | recent | 2026-08 Change to browse by: math math.CA References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from (opens in new tab) (opens in new tab) (opens in new tab)

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