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Further reduction of Poincaré-Dulac normal forms in symmetric systems. (English) Zbl 1165.34024

Consider in \(\mathbb{R}^n\) autonomous systems
\[ \dot{x}=f(x)=\sum_{k=0}^{\infty}f_k(x),\tag{1} \]
with \(f(0)=0\), where \(f\) is expanded around the origin as a power series with \(f_k(ax)=a^{k+1}f_k(x)\). The convergence of the \(f\) expansion is not considered, so all series are formal ones.
The Poincaré normalization procedure is based on a sequence of coordinate transformations generated by the solutions of homological equations. In the presence of resonances such solutions are not unique and one has to make an arbitrary choice for elements in the kernel of the relevant homological operator, different choices produce different high order effects. Different normal forms can be (formally) conjugated to the same system (1) and therefore (formally) conjugated among themselves. To reduce this redundancy in normal form classifications the authors discuss how a different prescription can lead to a further simplification of the resulting normal form in completely algorithmic way.

MSC:

34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34C14 Symmetries, invariants of ordinary differential equations
37G05 Normal forms for dynamical systems
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