Borel summability and Lindstedt series

TL;DR

This paper establishes conditions under which formal power series describing resonant motions in perturbed integrable systems can be summed using Borel summability, focusing on two-dimensional rotation vectors with Diophantine exponent 1.

Contribution

It provides new sufficient conditions for Borel summability of Lindstedt series in specific resonant cases, extending the understanding of formal series summation in dynamical systems.

Findings

Identifies conditions for Borel summability of Lindstedt series

Focuses on systems with two-dimensional rotation vectors and Diophantine exponent 1

Applies to resonant motions with formal power expansions

Abstract

Resonant motions of integrable systems subject to perturbations may continue to exist and to cover surfaces with parametric equations admitting a formal power expansion in the strength of the perturbation. Such series may be, sometimes, summed via suitable sum rules defining $C^\infty$ functions of the perturbation strength: here we find sufficient conditions for the Borel summability of their sums in the case of two-dimensional rotation vectors with Diophantine exponent $\tau=1$ (e. g. with ratio of the two independent frequencies equal to the golden mean).