The numerical search for the internal dynamics of NHIMs and their pictorial representation

TL;DR

This paper develops numerical methods to identify and analyze the internal dynamics of codimension 2 NHIMs in 3-degree-of-freedom Hamiltonian systems, illustrated through a magnetic dipole field example.

Contribution

It introduces strategies for numerically locating NHIMs and their manifolds, linking different approaches and providing visual representations of their internal dynamics.

Findings

Successful numerical identification of NHIMs in a magnetic dipole system

Comparison of different strategies for locating NHIMs

Visualization of NHIM internal dynamics and manifolds

Abstract

The topic of this article is the numerical search of codimension 2 Normally Hyperbolic Invariant Manifolds (NHIM) in Hamiltonian systems with 3 degrees of freedom and their internal dynamics. We point out relations between different strategies to find such surfaces numerically. We can start from index-1 saddles of the effective potential or from a partially integrable case and follow the NHIM along some curve in the parameter space of the system. Or, we can look for the stable and unstable manifolds of such surfaces by an appropriate indicator method. We show numerical examples for an electron moving in a perturbed magnetic dipole field.