The inner equation for generalized standard maps

TL;DR

This paper investigates solutions to the inner equation related to separatrix splitting in generalized standard maps, providing an explicit exponentially small expression and numerical evidence of its quantitative accuracy.

Contribution

It introduces an explicit exponentially small expression for solutions of the inner equation and demonstrates its effectiveness in predicting separatrix splitting.

Findings

Explicit exponentially small difference expression derived

Numerical evidence supports the inner equation's quantitative predictions

Inner equation provides insights even when the limit flow fails to do so

Abstract

We study particular solutions of the inner equation associated to the splitting of separatrices on generalized standard maps. An exponentially small complete expression for their difference is obtained. We also provide numerical evidence that the inner equation provides quantitative information of the splitting of separatrices even in the case when the limit flow does not.