Classification analytique des équations différentielles non linéaires résonnantes du premier ordre

Stokes Phenomenon and Confluence in Non-autonomous Hamiltonian Systems

“…, defined for x in the spiraling domain X ± ( ), ∈ E ± , of Definition 12 (for some 0 < η < π 2 arbitrarily small and some δ x , δ > 0 depending on η), and for u ∈ U, which brings Z H to its formal normal form (18). It is uniformly continuous on U × XE ± (25) and analytic on its interior.…”

“…Note that in our notation ± (u, x, 0) consists of a pair of sectoral transformations (u, x, 0) and (u, x, 0); it is a functional cochain using the terminology of [11,18].…”

“…A vertical infinitesimal symplectic symmetry, or shortly infinitesimal symmetry, of the normal form vector field Z G (18) is an analytic germ of a family of vector fields ξ(u, x, ) in the (u, x)-space that preserves:…”

A Galois D-groupoid for q-difference equations

“…For example, it was considered by G. Casale in [1] for a germ of diffeomorphism of (C, 0). He computed the list of possible D-groupoids on a disk, and establishing conditions, in each case, on the considered germ of diffeomorphism to be a solution of the D-groupoid, he recovered a part of the Martinet-Ramis analytic classification of germs of diffeomorphisms, as presented in [11].…”

Holomorphic germs and the problem of smooth conjugacy in a punctured neighborhood of the origin

“…Later, Martinet and Ramis (in [3]) showed that even a C 1 equivalence between (1.1) and (1.3) will imply either a holomorphic or antiholomorphic equivalence (although the statement and proof which they give is incomplete, it can be corrected easily). In the case that f (z) = z + z 2 , Ahern and Rosay [1] have proven that any germ of an entire function which is C 6 conjugate to f must in fact be holomorphically equivalent to f .…”