Confluence of singular points and nonlinear Stokes phenomenon (Q2761505)

The author studies a two-dimensional holomorphic vector field with an elementary degenerate singular point (saddle node) NEWLINE\[NEWLINE\dot{z}=z+O(|z|^2+|t|^{k+1}),\quad \dot{t}=t^{k+1}. \tag \(*\) NEWLINE\]NEWLINE There exists a formal (and in general divergent) power series \(\widehat{H}(z,t)\) such that the change \((z,t)\mapsto (y=\widehat{H}(z,t),t)\) brings system \((*)\) to its formal normal form NEWLINE\[NEWLINE\dot{y}=(1+\lambda t^k)y, \quad \dot{t}=t^{k+1}, \tag \(**\) NEWLINE\]NEWLINE which has a canonical first integral \(I(y,t)=yt^{-\lambda }\exp (1/kt^k)\). One can bring system \((*)\) to its normal form \((**)\) by holomorphic transformations in domains of the form \(U\times S\), where \(U\) is a neighbourhood of the origin on the \(z\)-line and \(S\) is a sector on the \(t\)-line centered at the origin. There exists a covering of a neighbourhood of the origin on the \(t\)-line by \(2k\) such sectors \(S_j\) and the corresponding holomorphic transformations \(H_j\) have the same Taylor series at \(0\) as \(\widehat{H}\) but it is impossible to glue them together into a transformation holomorphic at the origin of \(\mathbb{C}\times \mathbb{C}\). This is the nonlinear Stokes phenomenon. The impossibility is due to the Martinet-Ramis invariant (MRI), an obstruction arising from the orbital analytic classification of such vector fields. To define it set \(I_j=I\circ H_j\) and \(I_{j+1}=\varphi _j\circ I_j\). The MRI is the equivalence class of the system of functions \(\varphi _j\) w.r.t. cyclic shifts of the indices and transformations \(\varphi _j(t)\mapsto c\varphi _j(t/c)\) where \(c\in \mathbb{C}^*\) is independent of \(j\). For a generic deformation \((D*)\) of \((*)\) in which the singularity at \(0\) splits into nondegenerate linearizable singularities of \((D*)\), the MRI of \((*)\) is expressed as limit transition functions between the linearizing charts of the singularities of \((D*)\). It is proved that appropriate branches of the canonical first integrals of \((D*)\) converge to the sectorial canonical integrals of \((*)\). Similar assertions are proved for one-dimensional conformal maps having a fixed point with derivative \(1\) and for the Ecalle-Voronin moduli of their analytic classification, as well as for central manifolds of a multi-dimensional saddle node.