Normally Hyperbolic Invariant Cylinders Passing Through Multiple Resonance

TL;DR

This paper investigates the continuation of periodic orbits forming a smooth, normally hyperbolic invariant cylinder with holes, which is crucial for crossing multiple resonant points in classical systems.

Contribution

It introduces the concept of a normally hyperbolic invariant cylinder passing through multiple resonances, constructed from homoclinics and periodic orbits, advancing understanding of resonant crossings.

Findings

Construction of a $C^1$-smooth invariant cylinder with holes

Identification of the cylinder's role in crossing multiple resonances

Analysis of periodic orbits linked to homoclinic compounds

Abstract

We study the continuation of periodic orbits from various compound of homoclinics in classical system. Together with the homoclinics, the periodic orbits make up a $C^1$-smooth, normally hyperbolic invariant cylinder with holes. It plays a key role to cross multiple resonant point.