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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.

MSC:

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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