The analytic and formal normal form for the nilpotent singularity. (English) Zbl 1005.34034

Two-dimensional analytic vector fields with nilpotent singularity are treated by the authors. They prove that the Bogdanov-Takens prenormal form is analytic. They give also the complete formal orbital classification in a certain special case. As a corollary, they show that such a system with a centre has a local analytic first integral if and only if it is analytically orbitally equivalent to a certain Hamiltonian system. The proofs are very technical and difficult. The analyticity is proved by estimating the coefficients of the Taylor series expansions.


34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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