Convergence or generic divergence of Birkhoff normal form

2000/09/04 by Ricardo Perez-Marco, Ricardo Pérez-Marco, Perez-Marco, Ricardo
Computer Science · Mathematics · Physics and Astronomy · #34M35 #37J30 #37J40 #70K45 #Chaotic Dynamics (nlin.CD) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Functional Equations Stability Results #Mathematical Dynamics and Fractals #Polynomial and algebraic computation #math.DS #msc:34M35 #msc:37J30 #msc:37J40 #msc:70K45 #nlin.CD

paper · pdf · doi:10.48550/arxiv.math/0009028

arxiv created 2000/09/04 · openalex publication_date 2000/09/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that Birkhoff normal form of hamiltonian flows at a non-resonant singular point with given quadratic part are always convergent or generically divergent. The same result is proved for the normalization mapping and any formal first integral.

Citations

• Linearization of holomorphic germs with resonant linear part

Related

• Existence of divergent Birkhoff normal forms of Hamiltonian functions
• Convergence of the Birkhoff normal form sometimes implies convergence of a normalizing transformation
• Convergence versus integrability in Birkhoff normal form
• Normalization flow
• Integrability, hyperbolic flows and the Birkhoff normal form