Melnikov-Arnold integrals and optimal normal forms

TL;DR

This paper presents a new MA-integral based method to estimate secondary resonance sizes in Hamiltonian systems, simplifying calculations compared to traditional normalization techniques.

Contribution

It introduces a novel approach using Melnikov-Arnold integrals to efficiently estimate secondary resonance sizes without extensive normalization.

Findings

The MA-integral method accurately estimates secondary resonance sizes.

The approach applies to any order up to the optimal normal form.

It simplifies the analysis of Hamiltonian systems' resonance structures.

Abstract

The Melnikov-Arnold integrals (MA-integrals) is a well-known instrument used to measure the splitting of separatrices in Hamiltonian systems. In this article, we explore how calculation of MA-integrals can be used as well to estimate sizes of secondary resonances. Within the standard map model, we show how the newly developed MA-based procedure allows one to estimate the sizes of secondary resonances of any order (up to the order of the optimal normal form), without relying on the cumbersome traditional normalization procedure.