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Versal normal form for nonsemisimple singularities. (English) Zbl 1416.34031

Summary: The theory of versal normal form has been playing a role in normal form since the introduction of the concept by V. I. Arnol’d [“On matrices depending on parameters”, Russ. Math. Surv. 26, No. 2, 29–43 (1971; doi:10.1070/RM1971v026n02ABEH003827); “Lectures on bifurcations in versal families”, ibid. 27, No. 5, 54 (1972; doi:10.1070/RM1972v027n05ABEH001385)]. But there has been no systematic use of it that is in line with the semidirect character of the group of formal transformations on formal vector fields, that is, the linear part should be done completely first, before one computes the nonlinear terms. In this paper, we address this issue by giving a complete description of a first order calculation in the case of the two- and three-dimensional irreducible nilpotent cases, which is then followed up by an explicit almost symplectic calculation to find the transformation to versal normal form in a particular fluid dynamics problem and in the celestial mechanics \(L_4\) problem.

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

Software:

Fermat; FORM
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References:

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