Mould expansions for the saddle-node and resurgence monomials

Abstract

This article is an introduction to some aspects of Écalle's mould calculus, a powerful combinatorial tool which yields surprisingly explicit formulas for the normalising series attached to an analytic germ of singular vector field or of map. This is illustrated on the case of the saddle-node, a two-dimensional vector field which is formally conjugate to Euler's vector field $x^2\frac{\pa}{\pa x}+(x+y)\frac{\pa}{\pa y}$, and for which the formal normalisation is shown to be resurgent in $1/x$. Resurgence monomials adapted to alien calculus are also described as another application of mould calculus.

78 pages

Cited by in corpus (5)

• Normalization in Lie algebras via mould calculus and applications
• Initiation to mould calculus through the example of saddle-node singularities
• Hopf algebra techniques to handle dynamical systems and numerical integrators
• Logarithmic Derivatives and Generalized Dynkin Operators
• Ecalle's paralogarithmic resurgence monomials and effective synthesis