Symmetry-breaking for a restricted n-body problem in the Maxwell-ring configuration

TL;DR

This paper studies the dynamics of a massless body in the Maxwell-ring configuration, revealing symmetry-breaking and bifurcation phenomena through numerical continuation for n=7.

Contribution

It introduces a numerical continuation approach to analyze bifurcations and symmetry-breaking in the Maxwell-ring restricted n-body problem.

Findings

Identification of three equilibrium Zn-orbits for the massless body.

Construction of bifurcation diagrams showing Lyapunov and secondary bifurcations.

Observation of symmetry-breaking and period-doubling bifurcations.

Abstract

We investigate the motion of a massless body interacting with the Maxwell relative equilibrium, which consists of n bodies of equal mass at the vertices of a regular polygon that rotates around a central mass. The massless body has three equilibrium Zn-orbits from which families of Lyapunov orbits emerge. Numerical continuation of these families using a boundary value formulation is used to construct the bifurcation diagram for the case n=7, also including some secondary and tertiary bifurcating families. We observe symmetry-breaking bifurcations in this system, as well as certain period-doubling bifurcations.