Polynomial normal forms with exponentially small remainder for analytic vector fields.

*(English)*Zbl 1072.34039Authors’ abstract: A key tool in the study of the dynamics of vector fields near an equilibrium point is the theory of normal forms, invented by Poincaré, which gives simple forms to which a vector field can be reduced close to the equilibrium. In the class of formal vector-valued vector fields, the problem can be easily solved, whereas in the class of analytic vector fields, divergence of the power series giving the normalizing transformation generally occurs. Nevertheless, the study of the dynamics in a neighborhood of the origin can very often be carried out via a normalization up to finite order. This paper is devoted to the problem of optimal truncation of normal forms for analytic vector fields in. More precisely, we prove that for any vector field admitting the origin as a fixed-point with a semi-simple linearization, the order of the normal form can be optimized so that the remainder is exponentially small. We also give several examples of non-semi-simple linearization for which this result is still true.

Reviewer: Josef Hainzl (Freiburg)

##### MSC:

34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |

37G05 | Normal forms for dynamical systems |

##### Keywords:

analytic vector fields; normal forms; semisimple linearization; small divisors; optimal truncation; exponentially small remainders
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\textit{G. Iooss} and \textit{E. Lombardi}, J. Differ. Equations 212, No. 1, 1--61 (2005; Zbl 1072.34039)

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