On k-Summable Normal Forms of Vector Fields with One Zero Eigenvalue

• Peter De Maesschalck ^[1] ; Kristian Uldall Kristiansen ^[2]
  1. [1] University of Hasselt

     University of Hasselt

     Arrondissement Hasselt, Bélgica

     
  2. [2] Technical University of Denmark

     Technical University of Denmark

     Dinamarca

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 1, 2026
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • In this paper, we study normal forms of analytic saddle-nodes in with any Poincaré rank . The approach and the results generalize those of Bonckaert and De Maesschalck from 2008 that considered . In particular, we introduce a Banach convolutional algebra that is tailored to study differential equations in the Borel plane of order k. One of the subtleties that we take care of in this paper, is that nontrivial Jordan blocks are allowed in the linear part of the vector field. We anticipate that our approach can stimulate new research and be used to study different normal forms in future work.

• Referencias bibliográficas
  • Baldomá, I., Castejón, O., Seara, T.M.: Exponentially small heteroclinic breakdown in the generic hopf-zero singularity. J. Dyn. Diff. Equat....
  • Balser, W.: Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations. Springer (2000)
  • Balser, W., Braaksma, B.L.J., Ramis, J.-P., Sibuya, Y.: Multisummability of formal power series solutions of linear ordinary differential...
  • Bittmann, A.: Doubly-resonant saddle-nodes in (C3,0) and the fixed singularity at infinity in painlevé equations: analytic classification....
  • Bonckaert, P., De Maesschalck, P.: Gevrey normal forms of vector fields with one zero eigenvalue. J. Math. Anal. Appl. 344(1), 301–321 (2008)
  • Bonckaert, P., Verstringe, F.: Renormalization of planar analytic saddles. J. Dynam. Differ. Equ. 24(3), 569–593 (2012)
  • Braaksma, B., Stolovitch, L.: Small divisors and large multipliers. Annales De L’institut Fourier 57(2), 603–628 (2007)
  • Braaksma, B.L.J.: Multisummability of formal power-series solutions of nonlinear meromorphic differential-equations. Annales De L Institut...
  • Costin, O.: Asymptotics and Borel summability. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 141. CRC...
  • De Maesschalck, P., Kenens, K.: Gevrey asymptotic properties of slow manifolds. Nonlinearity 33(1), 341–387 (2020)
  • Loray, F.: Versal deformation of the analytic saddle-node. Asterisque 297, 167–187 (2004)
  • Malgrange, B., Ramis, J.P.: Multisummable functions. Annales De L Institut Fourier 42(1–2), 353–368 (1992)
  • Martinet, J., Ramis, J.-P.: Classification analytique des équations différentielles non linéaires résonnantes du premier ordre. Annales Scientifiques...
  • Put, M., Singer, M.F.: Galois Theory of Linear Differential Equations. Springer (2003)
  • Sauzin, D.: Nonlinear analysis with resurgent functions. Ann. Sci. Éc. Norm. Supér. (4) 48(3), 667–702 (2015)
  • Schäfke, R., Teyssier, L.: Analytic normal forms for convergent saddle-node vector fields. Annales De L’institut Fourier 65(3), 933–974 (2015)
  • Sibuya, Y.: Linear Differential Equations in the Complex Domain: Problems of Analytic Continuation. Translations of Mathematical Monographs,...
  • Stolovitch, L.: Classification analytique de champs de vecteurs 1-résonnants de (cn, 0). Asymptot. Anal. 12(2), 91–143 (1996)
  • Stolovitch, L.: Progress in normal form theory. Nonlinearity 22(7), R77–R99 (2009)