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

TL;DR

This work establishes that the formal center-manifold expansion $\phi(x)=\sum_{n=1}^{\infty} \phi_n x^n$ at planar analytic saddle-nodes has Gevrey-1 growth dictated by the analytic invariant $a$, with a precise asymptotic $(-1)^n\phi_n/\Gamma(n+a) \to S_\infty$. The authors develop a novel combination of a normal-form reduction, a refined Borel-Laplace approach, and a tailored Banach space to control the singularity at the Borel plane point $w=-1$, enabling a fixed-point construction and bootstrapping arguments that yield $|\phi_n| \le C\Gamma(n+a)$ and the convergence of $S_\infty=\sum (-1)^j p_j/\Gamma(j+a)$. They further apply the theory to a Riccati-family to locally characterize analytic center manifolds and provide numerical evidence for bifurcation curves in parameter space. The results connect the factorial-like growth of coefficients to the saddle-node invariants and open avenues for extending the framework to higher Poincaré ranks and to deeper questions about analyticity when $S_\infty=0$. Overall, the paper advances rigorous exponential-type asymptotics for nonlinear center manifolds via a robust Borel-Laplace methodology and bootstrapping, with implications for summability, resurgence, and invariant-manifold theory.

Abstract

In this paper, we show that the coefficients $φ_n$ of the formal series expansions $y=\sum_{n=1}^\infty φ_n x^n\in x\mathbb C[[x]]$ of center manifolds of planar analytic saddle-nodes grow like $Γ(n+a)$ (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 $φ_n$, 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.

Figures (3)

• Figure 1: Illustrations of the different domains $\Omega(\alpha,R)$ and $\omega(\alpha,r)$. Fig. (a) is in the "Borel plane" $w\in \mathbb C$, whereas Fig. (b) is in the $x$-domain.
• Figure 2: Illustration of the result in Proposition \ref{['proposition:riccati']}. In a neighborhood of the point (\ref{['eq:tr']}), $S_\infty^{-1}(0)$ is the union of the two intersecting branches in red and blue (corresponding to items \ref{['item1']} and \ref{['item2']} of Proposition \ref{['proposition:riccati']}).
• Figure 3: The sign of $S_\infty(a,b)$ determined by numerical computations (see the main text for details). Here $S_{\infty}>0$ is in yellow whereas $S_{\infty}<0$ is in green. The bifurcations $a\in (-\mathbb N)\setminus \{-1\}$, see Remark \ref{['remark:bif']}, are shown as purple discs.

Theorems & Definitions (39)

• Theorem 1.1
• Lemma 1.2
• proof
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Remark 1
• Lemma 3.1
• Remark 2