×

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

Citations:

Zbl 0279.58009
PDF BibTeX XML Cite
Full Text: Euclid