# Normalization in Lie algebras via mould calculus and applications

@article{Paul2016NormalizationIL, title={Normalization in Lie algebras via mould calculus and applications}, author={T. Paul and D. Sauzin}, journal={Regular and Chaotic Dynamics}, year={2016}, volume={22}, pages={616-649} }

We establish Écalle’s mould calculus in an abstract Lie-theoretic setting and use it to solve a normalization problem, which covers several formal normal form problems in the theory of dynamical systems. The mould formalism allows us to reduce the Lie-theoretic problem to a mould equation, the solutions of which are remarkably explicit and can be fully described by means of a gauge transformation group. The dynamical applications include the construction of Poincaré–Dulac formal normal forms… Expand

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#### References

SHOWING 1-10 OF 31 REFERENCES

Normalization in Banach scale Lie algebras via mould calculus and applications

- Mathematics, Physics
- 2016

We study a perturbative scheme for normalization problems involving resonances of the unperturbed situation, and therefore the necessity of a non-trivial normal form, in the general framework of… Expand

From dynamical systems to renormalization

- Mathematics, Physics
- 2013

In this paper we study logarithmic derivatives associated to derivations on completed graded Lie algebra, as well as the existence of inverses. These logarithmic derivatives, when invertible,… Expand

Formal differential equations and renormalization

- Mathematics
- 2016

The study of solutions of differential equations (analytic or formal) can often be reduced to a conjugacy problem, namely the conjugation of a given equation to a much simpler one, using… Expand

Mould expansions for the saddle-node and resurgence monomials

- Mathematics
- 2007

This article is an introduction to some aspects of \'Ecalle's mould calculus, a powerful combinatorial tool which yields surprisingly explicit formulas for the normalising series attached to an… Expand

Convergence of a quantum normal form and an exact quantization formula

- Mathematics, Physics
- 2011

Abstract The operator − i ℏ ω ⋅ ∇ on L 2 ( T l ) , quantizing the linear flow of diophantine frequencies ω = ( ω 1 , … , ω l ) over T l , l > 1 , is perturbed by the operator quantizing a function V… Expand

Six Lectures on Transseries, Analysable Functions and the Constructive Proof of Dulac’s Conjecture

- Mathematics
- 1993

The present paper gives a rapid, self-contained introduction to some new resummotion methods, which are noticeable for their high content in structure and revolve logically around the notions of… Expand

Normal Forms in Perturbation Theory

- Physics, Computer Science
- Encyclopedia of Complexity and Systems Science
- 2009

Normal form procedure This is the stepwise ‘simplification’ by changes of coordinates, of the Taylor series at an equilibrium point, or of similar series at periodic or quasi-periodic solutions.… Expand

Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems

- Mathematics
- 2003

Abstract. – We prove a theorem about the stability of action variables for Gevrey quasi-convex near-integrable Hamiltonian systems and construct in that context a system with an unstable orbit whose… Expand

Computing normal forms and formal invariants of dynamical systems by means of word series

- Mathematics
- 2015

We show how to use extended word series in the reduction of continuous and discrete dynamical systems to normal form and in the computation of formal invariants of motion in Hamiltonian systems. The… Expand

The Schrödinger equation and canonical perturbation theory

- Mathematics
- 1987

LetT0(ħ, ω)+εV be the Schrödinger operator corresponding to the classical HamiltonianH0(ω)+εV, whereH0(ω) is thed-dimensional harmonic oscillator with non-resonant frequencies ω=(ω1, ... , ωd) and… Expand