Bryuno Function and the Standard Map

TL;DR

This paper proves that invariant curves with Bryuno rotation numbers in the standard map are analytic for small perturbations, and establishes a relation between the convergence radius of the Lindstedt series and the Bryuno function.

Contribution

It demonstrates the analyticity of invariant curves under Bryuno conditions and links the Lindstedt series convergence radius to the Bryuno function.

Findings

Invariant curves are analytic for small perturbations with Bryuno rotation numbers.

The logarithm of the convergence radius plus twice the Bryuno function is uniformly bounded.

The relation between the Lindstedt series convergence and the Bryuno function is established.

Abstract

For the standard map the homotopically non-trivial invariant curves of rotation number satisfying the Bryuno condition are shown to be analytic in the perturbative parameter, provided the latter is small enough. The radius of convergence of the Lindstedt series -- sometimes called critical function of the standard map -- is studied and the relation with the Bryuno function is derived: the logarithm of the radius of convergence plus twice the Bryuno function is proved to be bounded (from below and from above) uniformily in the rotation number.