Improved Gevrey-1 estimates of formal series expansions of center manifolds

In this paper, we show that the coefficients ${\phi }_n$ of the formal series expansions $y={\sum }_{n=1}^{\mathrm{\infty }}{\phi }_n{x}^n\in x\mathbb{C}[[x]]$ of center manifolds of planar analytic saddle-nodes grow like $\mathrm{\Gamma }(n+a)$ (after rescaling $x$) as $n\to \mathrm{\infty }$. Here the quantity $a$ is the formal analytic invariant associated with the saddle node (following the work of J. Martinet and J.-P. Ramis). This growth property of ${\phi }_n$, which cannot be improved when the center manifold is nonanalytic, was recently (2024) described for a restricted class of nonlinearities by the present author in collaboration with P. Szmolyan. This joint work was in turn inspired by the work of Merle, Raphaël, Rodnianski, and Szeftel (2022), which described the growth of the coefficients for a system related to self-similar solutions of the compressible Euler. In the present paper, we combine the previous approaches with a Borel-Laplace approach. Specifically, we adapt the Banach norm of Bonckaert and De Maesschalck (2008) in order to capture the singularity in the complex plane.