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

Tables from this paper

Normalization in Banach scale Lie algebras via mould calculus and applications
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 ofExpand
Hopf algebra techniques to handle dynamical systems and numerical integrators
In a series of papers the present authors and their coworkers have developed a family of algebraic techniques to solve a number of problems in the theory of discrete or continuous dynamical systemsExpand
The Baker–Campbell–Hausdorff formula via mould calculus
The well-known Baker–Campbell–Hausdorff theorem in Lie theory says that the logarithm of a noncommutative product $$\text {e}^X \text {e}^Y$$eXeY can be expressed in terms of iterated commutators ofExpand
Rayleigh–Schrödinger series and Birkhoff decomposition
We derive new expressions for the Rayleigh–Schrödinger series describing the perturbation of eigenvalues of quantum Hamiltonians. The method, somehow close to the so-called dimensionalExpand
Time Dependent Quantum Perturbations Uniform in the Semiclassical Regime
We present a time dependent quantum perturbation result, uniform in the Planck constant for potential whose gradient is bounded a.e..We show also that the classical limit of the perturbed quantumExpand

References

SHOWING 1-10 OF 31 REFERENCES
Normalization in Banach scale Lie algebras via mould calculus and applications
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 ofExpand
From dynamical systems to renormalization
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
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, usingExpand
Mould expansions for the saddle-node and resurgence monomials
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 anExpand
Convergence of a quantum normal form and an exact quantization formula
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 VExpand
Six Lectures on Transseries, Analysable Functions and the Constructive Proof of Dulac’s Conjecture
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 ofExpand
Normal Forms in Perturbation Theory
  • H. Broer
  • 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
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 whoseExpand
Computing normal forms and formal invariants of dynamical systems by means of word series
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. TheExpand
The Schrödinger equation and canonical perturbation theory
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) andExpand
...
1
2
3
4
...