Improved Gevrey-1 Estimates of Formal Series Expansions of Center Manifolds

In this paper, we show that the coefficients ${\varphi }_n$ of the formal series expansions ${\sum }_{n=1}^{\infty }{\varphi }_n{x}^n\in xC\left[\left[x\right]\right]$ of center manifolds of planar analytic saddle-nodes grow like $\Gamma (n+a)$ (after rescaling $x$) as $n\to \infty$. Here, the quantity $a$ is the formal analytic invariant associated with the saddle-node (following the work of Martinet and Ramis). This growth property of ${\varphi }_n$, which is optimal, was recently (2024) described for a restricted class of nonlinearities by the present author in collaboration with 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. Finally, we apply the result to a family of Riccati equations and obtain a partial classification of the analytic center manifolds.