Finding how many isolating integrals of motion an orbit obeys

TL;DR

This paper evaluates the correlation dimension technique to determine the number of isolating integrals of motion in stellar orbits, optimizing parameters and validating results against established chaos indicators.

Contribution

It provides an analysis of the correlation dimension method's parameters, establishes optimal values, and validates its effectiveness in astrophysical potentials.

Findings

Optimal parameter values for the correlation dimension technique.

Validation of the method against SALI, Lyapunov exponents, and spectral dynamics.

Demonstrated reliability in estimating orbit integrals of motion.

Abstract

The correlation dimension, that is, the dimension obtained by computing the correlation function of pairs of points of a trajectory in phase space, is a numerical technique introduced in the field of nonlinear dynamics in order to compute the dimension of the manifold in which an orbit moves, without the need of knowing the actual equations of motion that give rise to the trajectory. This technique has been proposed in the past as a method to measure the dimension of stellar orbits in astronomical potentials, i.e., the number of isolating integrals of motion the orbits obey. Although the algorithm can in principle yield that number, some care has to be taken in order to obtain good results. We studied the relevant parameters of the technique, found their optimal values, and tested the validity of the method on a number of potentials previously studied in the literature, using the SALI, Lyapunov exponents and spectral dynamics as gauges.