The structure of phase space close to fixed points in a 4D symplectic map

TL;DR

This paper investigates the local structure of 4D symplectic maps near fixed points using visualization techniques, revealing patterns of stability that are consistent with galactic potential models and applicable to broader dynamical systems.

Contribution

It introduces a visualization method to analyze 4D phase space near fixed points and demonstrates the universality of stability patterns across different dynamical systems.

Findings

4D phase space near fixed points resembles surfaces of section near periodic orbits

The stability patterns are consistent with galactic potential models

The visualization method effectively reveals archetypical stability patterns

Abstract

We study the dynamics in the neighborhood of fixed points in a 4D symplectic map by means of the color and rotation method. We compare the results with the corresponding cases encountered in galactic type potentials and we find that they are in good agreement. The fact that the 4D phase space close to fixed points is similar to the 4D representations of the surfaces of section close to periodic orbits, indicates an archetypical 4D pattern for each kind of (in)stability, not only in 3D autonomous Hamiltonian systems with galactic type potentials but for a larger class of dynamical systems. This pattern is successfully visualized with the method we use in the paper.