Rotation numbers of invariant manifolds around unstable periodic orbits for the diamagnetic Kepler problem

TL;DR

This paper introduces a method to construct topological templates using symbolic dynamics for the diamagnetic Kepler problem, enabling accurate computation of rotation numbers of invariant manifolds around unstable periodic orbits.

Contribution

It presents a novel approach to determine rotation numbers via symbolic ordering, linking topological templates with phase space dynamics in the diamagnetic Kepler problem.

Findings

Rotation numbers match between topological templates and original definitions.

Symbolic ordering constrains the Poincaré section along stable manifolds.

The method confirms the topological template's effectiveness in analyzing invariant manifolds.

Abstract

In this paper, a method to construct topological template in terms of symbolic dynamics for the diamagnetic Kepler problem is proposed. To confirm the topological template, rotation numbers of invariant manifolds around unstable periodic orbits in a phase space are taken as an object of comparison. The rotation numbers are determined from the definition and connected with symbolic sequences encoding the periodic orbits in a reduced Poincar\'e section. Only symbolic codes with inverse ordering in the forward mapping can contribute to the rotation of invariant manifolds around the periodic orbits. By using symbolic ordering, the reduced Poincar\'e section is constricted along stable manifolds and a topological template, which preserves the ordering of forward sequences and can be used to extract the rotation numbers, is established. The rotation numbers computed from the topological template are the same as those computed from their original definition.