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Perturbative Expansions, Convergence of

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Mathematics of Complexity and Dynamical Systems

Article Outline

Glossary

Definition of the Subject

Introduction

Poincaré–Dulac Normal Forms

Convergence and Convergence Problems

Lie Algebra Arguments

NFIM and Sets of Analyticity

Hamiltonian Systems

Future Directions

Bibliography

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Abbreviations

Resonant eigenvalues:

Let B be a linear endomorphism of \({\mathbb{C}^n}\), with eigenvalues \({\lambda_1,\dots, \lambda_n}\) (counted according to multiplicity). One calls these eigenvalues resonant if there are integers \({d_j\geq 0}\), \({\sum d_j\geq 2}\), and some \({k\in\{1,\dots,n\}}\) such that

$$ d_1\lambda_1+\dots+d_n\lambda_n-\lambda_k=0\:. $$

If B is represented by a matrix in Jordan canonical then the associated vector monomial \({x_1^{d_1}\dots x_n^{d_n} e_k}\) will be called a  resonant monomial . (Here \({e_1,\dots,e_n}\) denote the standard basis, and the x i are the corresponding coordinates.)

Poincaré–Dulac normal form :

Let \({f=B+\dots}\) be a formal or analytic vector field about 0, and let \({B_\mathrm{s}}\) be the semisimple part of B. Then one says that f is in Poincaré–Dulac normal form (PDNF) if \({[B_\mathrm{s}, f]=0}\). An equivalent characterization, if B is in Jordan form, is to say that only resonant monomials occur in the series expansion.

Normalizing transformations and convergence:

A relatively straightforward argument shows that any formal vector field \({f=B+\dots}\) can be transformed to a formal vector field in PDNF via a formal power series transformation. But for analytic vector fields, the existence of a convergent transformation is not assured. There are two obstacles to convergence: First, the possible existence of small denominators (roughly, this means that the eigenvalues satisfy “near‐resonance conditions”); and second, “algebraic” obstructions due to the particular form of the normalized vector field.

Lie algebras of vector fields:

The vector space of analytic vector fields on an open subset U of \({{\mathbb C}^n}\), with the bracket \({[p, q]}\) defined by

$$ [p, q](x):=Dq(x)\,p(x) - Dp(x)\,q(x) $$

becomes a Lie algebra , as is well known. Mutatis mutandis, this also holds for local analytic and for formal vector fields. As noted previously, PDNF is most naturally defined via this Lie bracket. Moreover, the “algebraic” obstructions to convergence are most appropriately discussed within the Lie algebra framework.

Normal form on invariant manifolds:

While there may not exist a convergent transformation to PDNF for a given vector field f, one may have a convergent transformation to a “partially normalized” vector field, which admits a certain invariant manifold and is in PDNF when restricted to this manifold. This observation is of some practical importance.

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Authors and Affiliations

  1. Lehrstuhl A für Mathematik, RWTH Aachen, Aachen, Germany

    Sebastian Walcher

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  1. Sebastian Walcher
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Walcher, S. (2012). Perturbative Expansions, Convergence of. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_87

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