## Multidimensional formal Takens normal form.(English)Zbl 1190.34046

Consider an analytic germ of the form $$V=X+h.o.t.$$ where
$X=(n-1)x_2\partial_{x_1}+(n-2)x_3\partial_{x_2}+\dots+x_n\partial_{x_{n-1}}.$
The authors prove that the germ $$V$$ can be reduced by means of a formal change of the variables $$x_1,\dots,x_n$$ to the following form
$V^{\text{takens}}=X+F_1(G)\partial_{x_1}+\dots+F_n(G)\partial_{x_n},$
where $$F_j(G)=F_j(G_1,\dots,G_{n-1})$$ is a formal power series in $$G_2,\dots,G_{n-1}$$ with coefficients being Laurent polynomials in $$G_1=x_1$$. Moreover, the form $$V^{\text{takens}}$$ is unique in some sense. This result is a multidimensional analogue of the classical Takens normal form for a nilpotent singularity of a vector field [F. Takens, Publ. Math., Inst. Hautes Étud. Sci. 43, 47–100 (1973; Zbl 0279.58009)].

### MSC:

 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms 37C10 Dynamics induced by flows and semiflows

### Keywords:

nilpotent singularity; formal orbital normal

Zbl 0279.58009
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