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

TL;DR

This paper establishes precise Gevrey-1 estimates for the growth of coefficients in formal series expansions of center manifolds of planar saddle-nodes, linking growth rates to formal invariants and employing a Borel-Laplace method.

Contribution

It provides improved Gevrey-1 estimates for the coefficients of formal series of center manifolds, connecting growth to formal invariants and using a novel Borel-Laplace approach.

Findings

Coefficients grow like Gamma(n+a) after rescaling

Growth rate linked to formal analytic invariant a

Method combines previous approaches with Borel-Laplace analysis

Abstract

In this paper, we show that the coefficients $\phi_n$ of the formal series expansions $y=\sum_{n=1}^\infty \phi_n x^n\in x\mathbb C[[x]]$ of center manifolds of planar analytic saddle-nodes grow like $\Gamma(n+a)$ (after rescaling $x$) as $n\rightarrow \infty$. 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, Rapha\"{e}l, 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.