Initiation to mould calculus through the example of saddle-node singularities

David Sauzin (1)

1 IMCCE - Institut de Mécanique Céleste et de Calcul des Ephémérides

This article proposes an initiation to 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. This is illustrated on the case of saddle-node singularities, generated by two-dimensional vector fields which are 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 proves to be resurgent in~$1/x$.

Mots clés: Resurgence theory; mould calculus; saddle-node

Domaines: Systèmes dynamiques [math.DS]

David Sauzin. Initiation to mould calculus through the example of saddle-node singularities. Rev. Semin. Iberoam. Mat., 2008, 3 (V-VI), pp.147--160. ⟨hal-00201446v2⟩