Investigating the nature of motion in 3D perturbed elliptic oscillators displaying exact periodic orbits

TL;DR

This study explores the transition from regular to chaotic motion in a 3D perturbed elliptic oscillator potential, revealing that chaos rapidly increases with energy and demonstrating the effectiveness of the S(c) spectrum as a diagnostic tool.

Contribution

It introduces a method to predict 3D orbit behavior from 2D system results and validates the S(c) spectrum as a fast, reliable indicator of chaos in complex dynamical systems.

Findings

Chaos increases rapidly with energy in the 3D potential.

Approximately 97% of the phase space is chaotic at high energies.

The S(c) spectrum effectively distinguishes between regular and chaotic motion.

Abstract

We study the nature of motion in a 3D potential composed of perturbed elliptic oscillators. Our technique is to use the results obtained from the 2D potential in order to find the initial conditions generating regular or chaotic orbits in the 3D potential. Both 2D and 3D potentials display exact periodic orbits together with extended chaotic regions. Numerical experiments suggest, that the degree of chaos increases rapidly, as the energy of the test particle increases. About 97% of the phase plane of the 2D system is covered by chaotic orbits for large energies. The regular or chaotic character of the 2D orbits is checked using the S(c) dynamical spectrum, while for the 3D potential we use the S(c) spectrum, along with the P(f) spectral method. Comparison with other dynamical indicators shows that the S(c) spectrum gives fast and reliable information about the character of motion.