The Analytic and Formal Normal Form for the Nilpotent Singularity

Abstract

We study orbital normal forms for analytic planar vector fields with nilpotent singularity. We show that the Takens normal form is analytic. In the case of generalized cusp we present the complete formal orbital normal form; it contains functional moduli. We interprete the coefficients of these moduli in terms of the hidden holonomy group. © 2002 Elsevier Science (USA)

A preparation theorem for codimension-one foliations

“…In the case λ 2 = −λ 1 (including the nilpotent case λ i = 0), Theorem 4 was obtained by E. Stróżyna and H.Żo ladek [19]. They proved the convergence of an explicit iterative reduction process after long and technical estimates.…”

“…the normal form for the induced foliation) can be immediately derived just by setting f ≡ 1: coefficient g stands for the moduli of the foliation. The normal form (3) was also derived in [19].…”

Centers: their integrability and relations with the divergence

“…Teixeira and Yang [25] analyse the relationship between reversibility and the center-focus problem for systems (2) and (3), written in a convenient normal form. Strózyna and Zoladek [23] gave also a good normal form for nilpotent centers.…”

Local bifurcation of limit cycles and center problem for a class of quintic nilpotent systems

“…Hence, when 0 <h Consider the system 24) where g = {g 1 , g 2 , ..., g m-1 } is (m-1)-dimensional parameter vector. Let…”